G'day - YMMV as I am in Australia but here is my experience.
My pool is newly installed, but I have had a couple of pools over the years, all have had covers. These days in Australia, many localities require use of a cover, purely for water conservation against evaporation.
One thing is for sure, your evaporation water usage will be noticeably lower if you live in a dry hot area. That is the number one reason I have a cover. Where I live, with the cover off, I easily lose about 2cm of water a week, which relates to about 6 - 700 litres. With the cover on, maybe half a centimetre.
My new pool, part of the package included a "vented" solar pool cover. Not sure if these are available in US, but if they are, my personal opinion is don't bother about the vented selling point. On paper, it sounds really good, but in reality, I am unhappy and have my doubts about what it should do. I can elaborate if you want to know why.
The advice above about getting a thin one is probably a good idea and replace every xx amount of years. The thicker the cover, the heavier it will be to roll off and the roller will need to be stronger, to hold the weight (presuming you will use a roller). Thicker would be warmer water for longer.
Keeping leaves out, more likely yes than no, but in my case (dusty area) I have to deal with dust sitting on top and the leaves that don't blow off, you will need a plan to remove before removing the cover from pool. When you start to roll the cover off, everything will roll toward the opening and, the final loop on roller will cause everything to go into the pool. This can be particularly distressing with dust, but either way, without the cover, the dust would still get in, just not in one big dumping. You can minimise it by getting help and folding the end up as you pull the last bit out, but its not fail proof. This is particularly a problem for me, as the top of my cover is usually damp/wet (vented cover issue and issues with in floor cleaning - below).
If you don't plan to use a roller, you will need enough room around the pool to pull it out, otherwise you will need to fold it back on itself which will result in someone getting wet. The "bubble wrap" stuff on the water side will drag a bit of water out with it, unless you do it real slow allowing it to drain back into pool.
My pool is about 4 - 8 (Celsius) degrees warmer than when it is off, ambient temp and sunshine dependant and definitely stays warmer overnight. I can also turn my SWCG to a lower setting whilst cover is on.
My opinion is they look ugly when on the pool - due to dirt and dust on top.
If you have in floor cleaning with the rotating heads, I find that when the heads are pointing toward the wall of the pool, this can force water up over the edge of cover and it will flood the top of the cover. Same issue with solar pool heating, due to the drain back setup of that system, when it comes on, the pump has to clear the air from it's system which bubbles quite violently and throws water up onto the cover. This is my biggest grievance, one of which I am looking to solve - maybe a slightly larger cover to fold up on the edges slightly..
I had a kidney shaped pool years ago and just for ease, the cover was two piece. I think it was as a result of the size of the piece of "bubble wrap" that the generic cover came in, it was more economical to buy it as a piece and cut to shape. Jsut the way ot came, resulted in it being two pieces. It was about a 70/30 split and it wasn't too bad because when I just wanted a quick dip, I found I would only take one end out, which was easier.
In Australia's non tropical areas, I wouldn't be without a cover, purely to mitigate evaporation.
If I know rain is coming, I take it off. It's just a thing I do, but if you saw how much dust mine gathers, you'd probably do the same.
Hope that helps with your decision.
All the best.
