Aside from saving gas or electricity, why cover the pot?

It retains heat & steam pressure, so you achieve a simmer or even a boil with less power input & less water loss.

Great for long-cook sauces. Not so good for starchy things which may boil over if you're not careful, or have the pan too full.

As with most things, familiarity with how your stove/pan/lid combination reacts to a slow simmer will determine how well this works for you. If your stove won't drop low enough, you'll burn things, or need to get a simmer ring to reduce the risk. If you tend top make things too wet, you need to adjust over time.

Making anything to an online recipe needs you to be fully aware of how your own kitchen responds to wet/dry/timings. I don't think I ever blindly follow a recipe, I modify to how it needs to turn out based on my own experience with similar ingredients/quantities.
If I guess wrong, then I'm reducing sauce in a flat panic over the last 10 mins… not a good feeling, but familiarity breeds content ;))

There was at one time folk wisdom that said, "if it grows above ground, lid off, if it grows below ground, lid on"… which I have pretty much ignored for most of my adult life.
Sure, potatoes boil best with the lid on, as you obviously retain the heat, but I'm pretty sure I've never boiled a leafy vegetable in 20 years, I steam them… incidentally using the 'spare' steam from my lid-on potatoes.

Supplementing Tetsujin's answer, here are a few factors I've noticed:

Many sauces get rather splashy; putting the lid on means you don't need to clean the stove so much afterwards (and the wall behind, and the nearest cupboards...). Tomato-based dishes are particularly prone to this as they thicken, and the red can stain some materials.

With some dishes, as the liquid level drops, a layer of dried sauce forms on the sides of the pan. You can usually stir it back in, but not at the last minute, as it takes time to incorporate back into the dish. From starch-thickened sauces this can form a sheet that's quite hard to re-incorporate, but if the dish is tomato-based the dried bit mixes back in more easily. I even use that to my advantage when reducing passata (+herbs etc) for pizza sauce, but if it's undesirable the lid limits the drying out.

The lower power input needed can be enough (with a thick-based pan) to stop the sauce catching on the bottom if you don't stir it.

This is basically a question of thermodynamics -- you have a system in which there is heat being applied from the bottom, while being cooled from the top.

If there's a lid on the system, then you retain moisture in the space above the sauce, which reduces the amount of evaporative cooling. This means that you need apply less heat to the system, and it results in less of a temperature gradient through the sauce.

It's this temperature gradient that gets you into trouble -- a thick tomato sauce doesn't convect the way that a thin stock might, and so you'll get scorching (burning) on the bottom of the pot ... which can ruin your sauce.

As such, it's much more maintenance to cook the sauce with the lid off -- you have to stir it much more frequently. You can also end up with a slightly more "roasted" flavor to the sauce, as the higher heat at the bottom of the pot can caramelize the tomatoes depending on how often you're stirring it.

And there are things lost besides moisture and energy.

Volatile oils will be released into the air above the sauce, and if you have the lid off, they will go into the kitchen. This might make the kitchen smell great, but it can reduce the flavors in the dish being prepared. (although for some things, like cabbages, you intentionally cook lid-off so that the sulfur compounds don't stay in the dish)

And there's the cleanliness issue. Tomato sauce is notorious for spattering as it simmers. If you have a very tall pot that's not filled to the brim, this might not be an issue, but if your sauce level is near the top of the pot, and you've got the sauce at a simmer, the bubbles will burst spreading droplets of sauce all over your stove.