I run Flatpaks for gaming mainly because I want cutting edge Mesa (which is inherently used while running games from Flatpaks). It’s not necessary right now since Bookworm was just released and Mesa is not that old, but as a long-term plan I think this is more consistent. I don’t think Steam will have a problem keeping up to date without Flatpak, but Lutris is probably best kept cutting-edge via external Deb repo or Flatpak. In general Flatpaks are pretty terrific and I recommend using them for a lot of things nowadays anyway (I even used a lot of them on Arch Linux). There’s really no performance cost anymore, and most Flatpaks are configured seamlessly from the start so you don’t need to worry much about access issues.
Flatpaks can generally access anything they want, but you might have to allow that access manually, depending on how the Flatpak is initially configured (Flatseal makes this super easy). Usually this just means you’ll need to allow read-only or read/write access to specific directories where your personal files live. The biggest problem that Flatpaks have is that they absolutely can’t access /usr (and /usr/bin by extension), and this can make things like running mangohud %command% or strangle 60 %command% more or less impossible. You can actually still access stuff via /run/host/usr/bin from inside a Flatpak if you give it host permissions but things run this way usually get very confused and don’t work correctly. You can also try to use flatpak-spawn to do some interesting things from within the “sandbox”, but its usecases are limited in my experience. Notably you can cut off internet access from a game that’s being run, but flatpak-spawn also currently has a bug/oversight which wipes out the environment variables for the spawned process, which usually causes some havoc.
There are also “portals” that hook into Flatpak, which I don’t know nearly enough about. If you use KDE for example, when you “open” a file using a Flatpak’d program, KDE will swap in a native file dialog and let you pick a file from anywhere even if the Flatpak doesn’t have access, and then it grants a temporary one-time access to that file to the program. For example I disallow Discord from accessing practically everything, but I can still upload files from anywhere on my system to a Discord channel with the KDE file portal.
In terms of Proton versions and Wine versions etc, there’s no issues here. I usually use ProtonUp-Qt to install different versions, and Steam/Lutris/Bottles will have no issues accessing these even if they’re Flatpak’d.
Edit: Also, you can theme Flatpaks like this. You’ll need your system themes to be installed to ~/.themes and ~/.icons in order for Flatpak programs to access them, and then you can set default global overrides for the theming environment variables and access to ~/.themes etc. using Flatseal. This will allow all programs to use your themes without individual configuration.