I think I found a solution for now. Adding -vgui will launch the old Steam UI.

Yup, there are 2 main reasons I use Bottles.

1. I have only 1 bottle (wine prefix) and I put all of my games there. This means, I don't waste extra space for a separate wineprefix for every single game (I have a lot of games), and I don't have to set up a prefix every time I install a new one. I only had to install dependencies and set environment parameters once, and then every single game just works.
2. Bottles runs with Flatpak (sandboxed), which gives me an additional layer of security. We all know that downloading and running a pirated EXEs are not always safe.

Everything was working so well until this recent Steam update...

You probably missed it. I said there is no problem with the native Steam version, but I want to play my pirated games with my friends.

I don't understand what you're talking about. You don't have these problems? How are you running your Steam? Are you running the Windows version or the native Linux one?

Guys, I tried to add one of these games to Steam using the "Add Non-Steam game..." button. At first the game started, but for some reason it doesn't allow me to connect to the Xbox server, which is weird because it works with Bottles... The game I'm talking about is Grounded.

This might help. I will take a look. Thank you very much.

Man, Proton itself is just Wine + DXVK + VKD3D and maybe some minor tweaks. How would using Proton help my situation? I mean, if I was playing a legal copy of these games - yeah, running them directly on Steam would help, but these games are pirated copies, dude.

How would Lutris help me in this situation? I mean, isn't Lutris just a GUI for Wine, just like Bottles.