No, I wouldn't bother. I did that for some years before Proton became a thing but it was never great.

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.

Very edge case but Team Fortress 2 doesn't run in Proton mode (VAC error) but it works if you run Steam through Wine