You might want to check out the disAbility FAQ sticky. The second post there is about mobility issues.
It's possible to park an ECV to wait for/go on rides. A few rides can fit wheelchairs in the queue, but not ECVs so for those he'd either have to walk or transfer a wheelchair (the ride should have one) to go through anyway. It may be worth bringing the ECV through some of the longer queues, such as the Safari and Soarin which are a 1/4 mile long. There is a chance that a CM would use the manual override to move the ECV if it's blocking an area they need clear, but if you aim to leave in stroller parking areas, it shouldn't be a problem. If you do leave it outside a queue, you may want to bring either a poncho to put over it or towels that can be used to wipe it down in case of rain. Personally, I'd bring something to tie onto the ECV somewhere that stands out so you can find it easily if it gets moved (or you just plain forget where you put it). The WDW rentals do put your name on the front, but if you're looking at a whole bunch, it's going to be easier to pick out if there's a bright colored bandana or similar.
If you scroll down,
https://disneyworld.disney.go.com/guest-services/ecv-rentals/ has maps of where the
ECV rental locations are for each park. It's usually in the same general area as stroller rental.
As far as things specific to renting off-site, should you change your mind:
Personally, I found WDW buses to be very-doable with a mobility aid, but I use a wheelchair (manual or power) in my everyday life, anyway, including on public transportation. They're especially do-able if you go at off times. Since you sound flexible on what time you get to a park, that's worth keeping in consideration. Some (many/most?) drivers will offer to help you park an ECV.
If you were to rent a car, several of the companies people commonly mention here do rent ECVs that break down to fit in a car. I know for sure that Apple,
Buena Vista, and Walker do. I think Yellow Scooters does. The Pride Go-Gos and Pride Revos all break down for transport. I think that at least one of the rental companies also has a slightly larger ECV that can break down for transport. In general, if a company advertises "travel scooters", they'll have something that breaks down. Many drivers will offer to help park it on the bus. If you were really nervous, you could bring it on, set it so that the motors are disengaged (this is a lot easier than it sounds), and move it into the spot manually. It would be harder/take longer to break an ECV down and carry it on a bus than just loading one normally.
You do need to make sure you're within the weight limit and will be even with "stuff" on the
scooter. Apple's listed weight requirement have already subtracted the "stuff" weight so their weight limits as purely for the person (at least on the Go-Gos). The other companies list the factory supplied weight limit so you want to subtract 25-50lbs for the potential additional weight. And, of course, being below the limit means there's less strain on the battery so can make a difference on how long a full battery lasts. A full battery in good condition should last all day at WDW with a few precautions, but sometimes there are issues.