Hello @sysms, please copy the contents of Steam Runtime Diagnostics from Steam (Steam -> Help -> Steam Runtime Diagnostics) and put it in a gist, then include a link to the gist in this issue report.
@kisak-valve after completely rebooting my computer, the tray icon is now back again (just tried in LXDE). :/
I'm not sure, what has caused this phenomenon.
FWIW here is the link to the Steam Runtime Diagostics: https://gist.github.com/sysms/bc6a83a1adac180a4c4b5262552d1f00
Ok, so it just works only on the first start of Steam it seem.
After closing and reopening the Steam client, the tray icon is gone for good again.
I could narrow it down - it's not a Steam problem, as it seems.
If I stay in LXDE, the tray icon is available continuously, even after closing and restarting Steam.
If I switch once to a XFCE session, the tray icon is gone (in XFCE and also in LXDE after switching back to LXDE).
So I guess it's both DEs interfering somehow.
Nothing extracted yet.
Your system information
Steam client version (build number or date):

Distribution (e.g. Ubuntu): Debian 12
Opted into Steam client beta?: [No]
Have you checked for system updates?: [Yes]
Steam Logs: steam-logs.tar.gz
Please describe your issue in as much detail as possible:
Since the latest Steam stable client, there is no longer a tray icon displayed on Debian 12 and LXDE and XFCE.
This is very bad, since with one of the latest updates (presumably the 14.06.23) ALT-F4 started to minimize Steam into the tray (and not longer closing it).
So if I accidentally press ALT-F4 (since it's still in my muscle memory) to close steam, I now have to kill it from the command line ("killall steam") to exit it, and can no longer close/exit it from the tray icon.
Steps for reproducing this issue: