I know Steam on Linux uses a Wine environment to run Windows games, but I was recently reminded that you can run the Windows version of Steam in a Wine bottle. Is there any advantage to doing it this way instead of running the Linux client with Proton?
If you run steam on wine you lose the ability to run native Linux games and all the many proton patches that are not present in wine. Most of the games that run seamlessly on proton don't run so easy on wine. In proton each game gets their own winepath, own configs -many which are incompatible to each other - etc.
So.... Run native steam and run windows only games using proton or proton-ge in native steam .
Running the Linux version of Steam allows you to have different Wine environments for each game. I don't know how common it is, but it's happened at least a few times that some game would require a specific version of Proton to run.