Well you shouldn't be throwing unwanted toys away anyway, as there is the Rabbit Amnesty recylcing scheme. I have't seen anything saying they don't accept rechargeable toys, perhaps staff could clarify? Of course postage for you could become quite costly depending what service you use, but you could consider it an option for at least the rechargeable toys as that would likely only be a small packet.
You're seeing a lot of problems because of the information you're looking at. Most people don't look beyond how the toy performs, which is why you'll see a lot of positivity about them.
I don't know what battery type it is as I'm not technical, but you generally don't want your rechargeable toy to ever run flat. It can damage them. So I doubt it's the type that needs to be used constantly and emptied at every use. In an ideal world, you'd have your rechargeable toy on charge very frequently. For a lot of people this is after every use. The advice is generally to charge them all at least every 3 or 6 months (I can't remember which, but I think it is 6?) regardless of how much use they have had, so again maybe that will tell you what type of battery they use?
You also shouldn't be using rechargeable batteries in battery powered sex toys as it can damage them. So the comparison cost and performance wise would be between non rechargeable batteries and rechargeable toys.
Here in the UK, the cheap batteries last bugger all time in a sex toy. In fact, a lot of the time you have to change them every time you want to play with the toy if you want to have it on full power. The more expensive batteries like duracell last longer, but depending on the toy you may still only get a couple of uses before the toy starts running flat. For people who like having their toys run at full power everytime, this makes rechargeable sex toys the clear choice as they don't lose power like that, they tend to run full strength every time then simply cut out when they are out of charge.
Power level wise, in my experience rechargeable toys are always more powerful than battery operated alternatives. They tend to be deeper and more rumbly as well, whereas battery toys would feel buzzy and shallow. Most people have a preference for type of vibration, and often 'power queens' will go for the deeper, rumbly vibrations.
A lot of people like them for the fact that they are more powerful alone, and this is about as far as they look into them. That's what I mean by you seeing more negatives because you are looking further, there are lots of people who are sold by nothing more than a more powerful toy that they can pick up and play with without bothering to find batteries. That's all they need to hear to think they are superior.
As a comparison, I have a rabbit that takes 4 AA batteries (and yes, I think you're correct that more/bigger batteries means more power). It's probably my most powerful battery toy. I also have a rechargeable rabbit. The latter blows the first one out of the water in terms of power. The difference really is staggering. The rechargeable toy is deeper, more rumbly (like a mains powered toy would be) and even on the first setting is so much more powerful than the battery rabbit. Quite often people notice this difference and forget about anything else!
You're perfectly correct that they could run out of charge in the middle of sex. However, the same could be said for battery toys. It's all about managing charging your toy, many rechargeables last 2hrs and sometimes even longer on one charge. So it's a simple case of popping it on charge before use or remembering to charge it regularly, which really isn't any more hassle than digging out a fresh set of batteries or remembering to buy new ones.
Power consumption, I have no idea. Being USB charged I can't imagine it is much. I know I don't notice any difference in my electric use when I have a 'charge the whole toy box' week, so it can't be much. Again, perhaps a staff member would have this information?
I don't think you're overlooking anything, you're just looking more into them than what most people do :)