If you really are going to only play 360 games... why not just get a used 360? It'll be less than half the price of a used XBO and will have 100% compatibility unlike the XBO. Unless you get an expensive XBO X, which it doesn't sound like you're going to, the performance on XBO and 360 should be similar. It depends on a game by game basis, some perform better some may even perform worse.
I finished fallout 3 on XBO just recently, I imagine it runs the same as it did on 360, 30fps with drops. The game completely froze with an awful buzzing sound and had to be rebooted about 3-4 times but in a 70 hour playthrough it's not the end of the world, no idea if those crashes are a symptom of BC or if they occurred on 360 too.
"Do I get to play them on 1080p on Xbox One and get solid 30fps?" - Very few games on the 360 were 1080p and the FPS on the XBO may be better, the same or worse it's a case by case basis.
The 360 did have less games with performance issues than the PS3 (Bayonetta, GTA IV) but in most cases it's not going to be a game changer.
Loading times I believe should also be the same on XBO and 360. Fallout 3 didn't feel faster anyway.
LTD: PS4 - 125m, Switch - 110m, XBO - 51m
2020: PS4 - 9m, Switch - 22.5m, XBO - 2.5m, PS5 - 4.5m, XBX - 2.8m