I'll play, even though it looks like OP has already made their decision...
Ha. My kids like Animal Kingdom (we have not stayed there with them, but used to go often to Sanaa and Boma. Their favorite resort (now 8 and 4) is Grand Floridian! I asked my 8yo and she likes the water slide and pool play area, and the restaurants (specifically, 1900 Park Fare).
As others have said, every 3 years is difficult with points. Every other year is doable. And in the short term, you can stay in studios in VGF for a few trips. August you should have a fairly easy time switching resorts, but Christmas is very hard (and very points hungry). A week in a studio at VGF at Christmas is around 200 points.
(I will add, for your age kids, if you keep the twins in cribs for a while, and your oldest sleeps in the single murphy, a studio can feel pretty spacious because you don't have the couch / queen murphy pulled down. We did this for several trips until our youngest needed to sleep in a bed.)
I also love BLT *but* studios there only sleep 4 once the littles are out of cribs. Have stayed in a BLT studio once with DH for a long weekend, and there is a big difference between the studio and the 1br there.
We used to rent a double Citi Mini stroller and one time we were staying at the Swan and the kids had fallen asleep at
MNSSHP (the halloween party). We kept them in the stroller as long as possible, then I carried our younger one, and we woke the older one to get on the bus, where she slept again in the seat. Luckily the bus wasn't took crowded, but it was a fair bit of carrying. When my kids were younger, I used to also bring a boba (baby carrier) because they could sleep on the bus in those. Although post-pandemic, they are probably going to enforce more social distancing, there have been times in the past when DH and I were both standing - I had a toddler/baby in the boba, he was managing a double citi mini, and my older kid was given a seat. It's not easy.
As for Minnie Vans - yes - they provide seating only for 2 in car seats/boosters. But again, our last trip Xmas/New Years 2019, it was very hard to get a Minnie Van at busy times (park open, and dinner).
Renting would also be a good option now, as many owners are putting their points up for rent and you can try many different resorts before buying. I love AKV/AKL the resort, but wasn't a huge fan of the room decor. I thought I would dislike VGF because it seemed sort of stuffy, but ended up loving it there, even before the walkway news. If we had tried a BLT studio before buying, we probably wouldn't have bought BLT, or we'd have started with enough points to stay in 1BR or larger (we also decided to start small, and started adding on only months later). Also, resale prices will probably continue to go down.