i have rented two scooters from disney each of the last three times i took my parents. they were 77, 80 and 81 on those trips. walking the parks was not an option.
we chose not to rent from an outiside source for several reasons. we rent a car, we find it a faster way to get where we're going with no lines to stand in for busses, etc. and also did not want to deal with getting the ECVs on and off the disney busses and boats, etc. (we often stay poly/wilderness lodge etc where boat is your fastest transportation.)
we did not want to have to disassemble and reassemble two carts each time we went somewhere in the car. not to mention lifting them in and out of the car. so decided getting them in the parks was easier.
also, a few times the carts broke down (ok, well, maybe my mom hit a wall or something and the auto-stop thing kicked in, once just unknown reason it wouldn't go), the advantage of onsite rental? the CM told us to just go into the ride we are planning to go to, they took care of getting it changed out, transferred over the stuff in the baskets, and we were ready to go when we came back out! now that's service you can't get from an outside source.
all rental places are indeed inside the turnstile/entrance. so it does take about 10-15 min after you are through gates and trying to get to where you are going. same thing at end of day- about 5-10 min to do paperwork. this was not a big deal, as most places let you inside the park before you can actually go to the rides, and hold you in a main area to wait. so we were able to rent them and still keep up with crowds when they drop the rope for THE TOY STORY MIDWAY MANIA MARATHON !!
downside of renting outside? you only have them IN the parks. but all hotels have wheelchairs that you can use while on property, so we would drop the parents off at the front door and i would go get their wheelchairs from bell services, while someone else parked the car. (although if your parents have handicapped parking stickers, valet parking is free and you can just all get out at the front door and go in (since we have young healthy relatives with us, we just felt guilty doing that). we could get them all around the property easily with this wheelchair- to pools, restaurants, gift shops, etc.
not knocking off site rentals, but we very much prefer onsite rental, and it is a minor inconvenience.
as far as your folks running over other folks, yeah, it's going to happen. folks just right in front of you to beat you to where you're going.
but they dont' go very fast, so no major hospital requirng injuries reported. but lots of apologies! they do take some getting used to, and it does take longer to get through crowded areas. but in non crowded areas, i couldn't keep up with THEM- i had to run to stay with them, so it's a fair trade off.