4 weeks after DarkRacoon, and I have the same problem. Steam kind of launches, shows up like an invisble window, then closes again. I also need to kill the Steam process, uninstall through Software Manager and then re-install again. All games are still installed, so it's just annoying to re-install Steam itself. And btw, yes I have ran the dpkg apt update mentioned in the Details section.
Because I was a totally new Linux user fresh from Windows I will be as exact and clear as possible for anyone who also has the same first experience as me: The Software manager installs this client, the client must be run separately to install Steam. It will not automatically open and install Steam without user say so. (Aside from that 3 seconds of confusion working that out my experience has been that it works smoothly and easily. Just don't forget to go into Steam> settings> compatibility, in the Steam client itself and check/turn on compatibility mode for all games as this wasn't on by default when it first loaded up for me.)
This application does not work properly for me. After restarting the system, the program does not want to start and quits and runs forever. It is difficult for me to explain it in words. The only way to "solve" this problem is to uninstall the application and reinstall it.
I find this client to be satisfactory for the most part. Things such as that I cannot snap the client into place like other programs or the occasional stutter and freeze dampen my experience with it. It does work however and is used on my system on a daily basis! Tested on Linux Mint 22.