Yes, you can keep an everyday useful kitchen knife serviceable with the Anysharp you linked. For the majority of non-serrated knives sold in a supermarket it will be fine. I've got one, and use it on my Ikea, Sainsburys and Victorinox knives. I can get a better edge with stones, but only once I've got my hand back in, so I don't bother. I've got some better knives that it doesn't work on because the angle is different - so I end up not using those as much. It's a bit aggressive really, so probably wouldn't be ideal on a quality knife even of the right angle, except for restoring a chipped blade. It's not quite foolproof, but nearly. The suction cup needs a good surface on something heavy to stick to, end you need to be careful to hold the blade square to the grinders.

That steel is OK for finishing the job if you can be bothered, but it's optional (I've got almost the same one, and only really use it before carving meat, which I do about once a year).

This will seems almost heretical to some people here, but if you just want to cook and eat, and not play with knives, that's fine. But in that case don't waste your money on posh knives either.

Let's define three variables with which we want to characterize knife sharpening tools:

1. The best possible end state of the knife if the tool has been used expertly
2. The worst possible end state of the knife if the tool has been used improperly
3. The amount of effort needed for a person to go from "have never used it" to "using it expertly"

The first thing you have to recognize is that these three variables are independent of each other. The second one is that the evaluation doesn't somehow reverse for different kinds of knife - it is not that the type of knife doesn't matter, because an expert with a whetstone will get a worse "best possible result" with a cheap stamped knife than with a quality knife. It is that, if tool A performs better than tool B in one of the three dimensions on a quality knife, it also performs bettre in one of the three dimensions on a bad knife (there are exceptions, but I don't want to go into deep details here).

Now imagine a three-dimensional space spanned by the three variables (I don't have a way to draw it easily for this answer). It is important to note that it is not a single quadrant - so if the initial state of the knife is at zero, then the worst possible state for some tools can get into negative numbers.

The traditional system of a sharpening stone (or a combination of sharpening stones with different grits) and a leather strop for honing is at one corner of this space: It has a higher "best possible result if expertly used" than all other tools, a worse "worst possible result" than all other tools, and a higher "effort to learn" than all other tools. This makes it the best choice for afficionados - they happily expand the effort to learn and stop caring about the "worst possible result" variable.

If you are looking at the other tools, the problem is: they are all over that space, and you have no way to reliably find out where they are. Sure, it would be simple to use a tool which either has a very small gap between "best possible result" and "worst possible result" and as long as both are above zero, this would be OK for you. Or finding a tool which has a very low "effort to learn expertly" and an acceptable "best possible result". In both cases, you will save effort as compared to learning a whetstone. But simple is not easy: since you cannot recognize which tool fulfills these criteria, you are left with trying them out.

You can try to reduce the uncertainty by the usual way of searching reviews for a specific make and model. The problem here is that reviews usually don't recognize these three variables, but instead tend to give a single evaluation of the tool's performance, which is neither the best nor the worst possible end result, but the end result obtained with the reviewer's skill, and we cannot know what the reviewer's skill is. This makes their reported result a bad predictor for your future result if you buy the tool. Similarly, the price of the tool is also not a good predictor.

So, if you are OK with the uncertainty, you can buy a random (or, if you prefer, a well-reviewed) tool, try learning to use it, and if you are happy with it, stay with it. If you arrive at a point where the result is not satisfying for you, and you are either a) sure that the tool cannot do better, or b) you are not willing to invest more time in learning this tool, you switch to the next tool.

If certainty is more important to you, you can still go back to that whetstone, and invest the time needed to learn it until your knives come out sharper than you started. This is the same process as learning to do it perfectly, you will just experience success earlier, because you have more modest requirements.

What we can't do is to point to a specific sharpener tool and tell you "this will work well for you". First, I doubt that any of us has systematically researched different tools in all three dimensions to be able to tell where they fall on them, and second, even if somebody did, our site doesn't do specific brand/model recommendations. If you think reviews will be informative for you, you will have to look them up somewhere else, for example in test magazines or customer feedback in online stores.

Now to say a couple of words directly about things you stated in comments:

My impression was that for some knives, there is no option that does not require great skill but that for more everyday kitchen knives it is possible to get a reasonable result with something resembling one of the products I linked.

No, there is no such connection. Typically, if a tool gets a cheap dull knife slightly sharp, it will also get a good quality dull knife slightly sharp (again, with exceptions). It is just that "slightly sharp" tends to be unacceptable for people who buy the best knives, so they may express the opinion that it makes no sense to use these tools on their knives.

but then why are those products sold in supermarkets?

They are being sold because there are people who buy them, period. And all the ones get bought, both the good and the bad ones, because as explained before, as a buyer you cannot know which one is good and which ones is bad. When I said that these tools are all over the three-dimensional evaluation space, I meant it - some of them will always turn a dull knife into a duller knife, and no amount of practicing will change that. So, don't expect that just because there is a market for a tool (or even positive reviews) that it is good.

Based on the tests conducted by America’s Test Kitchen, an electric knife sharpener is probably the most effective and skill-wise the least demanding option.

The "best" results will be with

…a set of whetstones, a steel & a strop.
This will take you time to learn, which is specifically excluded in the question premise.
… or a guided belt system.
This would be stupidly expensive.
…or a hand-held guided system.
This will be much easier to learn, but still requires somewhat more dedication that I feel you want to apply to the task.

The second best result will be with an all-in-one electric grinding system - a drag-through grinding wheel system not a belt-driven one. With one of these you cannot get the angle wrong; 2 or 3 grades of wheel can get your knives sharp, then keep them sharp without rapidly grinding the metal away. A good one, however, is still going to be a hundred bucks-ish. [A cheap one is a total waste of money - I had one, it was pants.]. My new one cost the best part of 200 [I had to pay the premium for a US one made for UK 240v] Yes, it's expensive, yes, it's fabulous, but yes I decided that's what I would go for after many years of trying to use other systems, with inadequate skill or inadequate results.
This is the one I ended up with - Chef'sChoice Trizor XV - which is specifically to bring EU 20° knives to 15° so you don't necessarily need one that posh;) they do cheaper 2-wheel 20° versions.

After that would be one of the myriad static drag-through systems like your AnySharp. This will, quite aggressively, rip a new edge onto just about anything you pull through it.
They are not subtle, but they will put a 20° edge on a knife.
They will, over time, start to produce a wavy edge on your knife. They will also really take off quite a bit of the blade's hardened edge each time. If you are using relatively cheap supermarket knives & not trying to make delicate sushi, then you may probably never care.

The edge this provides will also be quite short-lived, because there's no honing aspect. You can get drag-through honers, which you can use daily/weekly until honing no longer gives you your edge back - then you go back to the AnySharp one more time.
With a drag-through honer you need to be aware of what blade type they are made for. Japanese knifes & such as Kitchen Devil with skinny flexible blades are set at 15°. Regular European & US blades are set at 20°, so though one of those dinky little roller-wheel Kitchen Devil sharpeners would look the best deal, they're wrong for your AnySharp 20° edge.

BBC Good Food has a few alternatives of the spinny-wheel variety. I can't say I've ever had one I'd recommend as a "sharpener" but as a honer, maybe so.
There are also twin-slot pull-through systems, such as this Wüsthof which might just do the job of your AnySharp plus a honer, in one package.

So, TL:DR
Fancy electric, a dual pull-through system like the Wüsthof, or AnySharp plus a honer would be my choices, in descending order.

Quick guide to why you hone & what honing is - What's wrong with if you only sharpen, and never hone, your knives?

Buy new knives.

There’s a very simple solution for having sharp, relatively cheap knives in your kitchen without having to learn any new skills or buy complicated, specialised tools: whenever your knives start to get dull, buy new knives!

the simplest thing I can do to make my knives better at cutting, preferably with equipment that's not too expensive and is easy to get hold of

Get a $10 honing steel and use it on properly-sharpened knives. It's simple to use, will dramatically stretch out the time between sharpenings and will give the best results while extending the life of your knives. There's a reason you see Iron Chefs like Zakarian and Norimoto constantly honing their knives during timed cooking competitions: the best results in less time!

I use my knives a lot more than most home cooks. I was tired of, frustrated by and injured using knives that didn't cut properly. I learned from web sites how to sharpen knives with a cheap set of water stones - contrary to popular belief, it wasn't difficult. But the real revelation was that frequent use of a honing steel brings the edges back to "razor sharpness" without the time-consuming and/or expensive sharpening that removes so much material from the blade. The "secret" is that the honing steel orients the material at the very edge of the blade, turning it into a micro-serrated cutting wonder.

A professional cook gave me a valuable insight: after honing with the steel, the edge is more effective on the pull stroke than on the push stroke. With a honing steel and this simple tip, paper-thin slices of a ripe, juicy tomato are flawless.