What is it that a filter Coffee maker can do that we can't do in a better or equivalent way at home?

The method outlined in the linked page is closest to French press brewing. You also have perfect control of the of the temperature of the water, and the length of the brew time.

A French press works essentially the same way, but the plunger filters out the grinds by pressing them to the bottom of the pot. If you pour immediately, the outcome is essentially the same. If you don't pour the coffee out right away, some people believe that it picks up off flavors from the grounds at the bottom of the pot. This method also produces coffee with a certain amount of grounds or sandiness at the bottom.

I have never heard of an automatic coffee maker that fully replicates the so-called straining method from that page.

After the French press, perhaps the most similar is the manual drip coffee maker, such as the famous Melita cone drip coffee maker or the Chemex. Normally, with this type of equipment, you put the grounds in the filter in the cone, over the coffee pot. You then moisten them, and slowly pour hot water through. It seeps through the grounds, picking up flavor, and the drips into the pot. The advantage of this is that you control the water temperature and pour time, which some coffee aficionados are very particular about in getting their perfect cup.

An automatic drip coffee maker essentially automates the manual drip process, at the cost of giving up control of the temperature and pour rate to the machine. The manufacturers do try to set them close to what are often considered ideal values. They are pretty successful for pour rate, but only the best machines do as well on temperature.

What you gain is a high level of convenience, which many people find an acceptable trade off. These are probably the most common type of coffee maker in the world, due to their simplicity, effectiveness, and convenience.

Please note that coffee generates a lot of passion and excitement among its fans. I believe the truth is that with quality coffee, and care, and reasonably good equipment, any of the brewing methods can produce a quality cup, although each of us may have our favorites.

If you like the so-called strainer method from that web page which is specifically about when you don't have specialized coffee equipment, I suggest you try a French press. It is not automatic, but it can produce a very high quality result.