I was getting the exact same error on Debian 9.3.0 (Stretch), using nvidia-drivers from "stable".
Had to resort to a backup from before the Dec 15 update to make it work again.

Found a workaround. The steam client is actually running, but the GUI is hidden. It can be shown by using the system-tray icon, but this is hidden in Gnome 3.26.2 as the legacy system-tray was removed. You can get it back by using extensions like TopIcons Plus:
https://extensions.gnome.org/extension/1031/topicons/

I'm running Xmonad as WM without any tray.

When I start the updated steam, the following happens:

cbm80@castle:~$ steam
Running Steam on debian 9 64-bit
STEAM_RUNTIME is enabled automatically
Pins up-to-date!
[2017-12-16 20:23:05] Startup - updater built Oct 30 2017 19:09:32
[2017-12-16 20:23:05] uninstalled manifest found in /home/cbm80/.steam/package/steam_client_ubuntu12 (1).
[2017-12-16 20:23:05] Found pending update
[2017-12-16 20:23:05] Installing update...
[2017-12-16 20:23:05] Extracting package...
[2017-12-16 20:23:13] Installing update...
[2017-12-16 20:23:13] Cleaning up...
[2017-12-16 20:23:13] Update complete, launching Steam...
[2017-12-16 20:23:13] Shutdown
Restarting Steam by request...
Running Steam on debian 9 64-bit
STEAM_RUNTIME has been set by the user to: /home/cbm80/.steam/ubuntu12_32/steam-runtime
Found newer runtime version for 32-bit libdbus-1.so.3. Host: 3.14.14 Runtime: 3.5.8
Found newer runtime version for 32-bit libgtk-x11-2.0.so.0. Host: 0.2400.31 Runtime: 0.2400.10
Found newer runtime version for 32-bit libpangox-1.0.so.0. Host: 0.0.0 Runtime: 0.3000.0
Found newer runtime version for 32-bit libcurl.so.4. Host: 4.1.0 Runtime: 4.2.0
Found newer runtime version for 32-bit libvulkan.so.1. Host: 1.0.39 Runtime: 1.0.54
Found newer runtime version for 32-bit libGLU.so.1. Host: 1.3.1 Runtime: 1.3.8004
Found newer runtime version for 64-bit libpangox-1.0.so.0. Host: 0.0.0 Runtime: 0.3000.0
Found newer runtime version for 64-bit libdbusmenu-glib.so.4. Host: 4.0.12 Runtime: 4.0.13
Found newer runtime version for 64-bit libSDL2-2.0.so.0. Host: 0.4.1 Runtime: 0.5.0
Found newer runtime version for 64-bit libvulkan.so.1. Host: 1.0.39 Runtime: 1.0.54
Found newer runtime version for 64-bit libGLU.so.1. Host: 1.3.1 Runtime: 1.3.8004
[2017-12-16 20:23:29] Startup - updater built Dec 15 2017 11:47:44
[2017-12-16 20:23:29] Verifying installation...
[2017-12-16 20:23:30] Verification complete
Refresh rate: 0
Refresh rate: 0
Refresh rate: 0
Refresh rate: 0
Refresh rate: 0
Refresh rate: 0
Refresh rate: 0
Refresh rate: 0
Refresh rate: 0

and then the process exits. There's no process named steam anywhere to be found.
Running with gdb reveals:

cbm80@castle:~$ DEBUGGER=gdb steam
Running Steam on debian 9 64-bit
STEAM_RUNTIME is enabled automatically
Pins up-to-date!
GNU gdb (Debian 7.12-6) 7.12.0.20161007-git
Copyright (C) 2016 Free Software Foundation, Inc.
License GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>
This is free software: you are free to change and redistribute it.
There is NO WARRANTY, to the extent permitted by law.  Type "show copying"
and "show warranty" for details.
This GDB was configured as "x86_64-linux-gnu".
Type "show configuration" for configuration details.
For bug reporting instructions, please see:
<http://www.gnu.org/software/gdb/bugs/>.
Find the GDB manual and other documentation resources online at:
<http://www.gnu.org/software/gdb/documentation/>.
For help, type "help".
Type "apropos word" to search for commands related to "word"...
Reading symbols from /home/cbm80/.steam/ubuntu12_32/steam...(no debugging symbols found)...done.
(gdb) run
Starting program: /home/cbm80/.steam/ubuntu12_32/steam 
[Thread debugging using libthread_db enabled]
Using host libthread_db library "/lib/x86_64-linux-gnu/libthread_db.so.1".
[2017-12-16 20:32:57] Startup - updater built Dec 15 2017 11:47:44
Looks like steam didn't shutdown cleanly, scheduling immediate update check
[New Thread 0xf4c10b40 (LWP 25796)]
[2017-12-16 20:32:57] Checking for update on startup
[2017-12-16 20:32:57] Checking for available updates...
[New Thread 0xf49ffb40 (LWP 25797)]
[2017-12-16 20:32:58] Download skipped: /client/steam_client_ubuntu12 version 1513371133, installed version 1513371133
[2017-12-16 20:32:58] Nothing to do
[2017-12-16 20:32:58] Verifying installation...
[2017-12-16 20:32:58] Performing checksum verification of executable files
[2017-12-16 20:32:59] Verification complete
[New Thread 0xef6c9b40 (LWP 25830)]
[New Thread 0xed7a2b40 (LWP 25860)]
[New Thread 0xecdffb40 (LWP 25861)]
[New Thread 0xecaffb40 (LWP 25862)]
[New Thread 0xec7ffb40 (LWP 25865)]
[New Thread 0xec4ffb40 (LWP 25866)]
[New Thread 0xec3feb40 (LWP 25867)]
[Thread 0xecaffb40 (LWP 25862) exited]
[New Thread 0xeb853b40 (LWP 25868)]
[New Thread 0xeafffb40 (LWP 25869)]
[New Thread 0xeaefeb40 (LWP 25870)]
[New Thread 0xeadfdb40 (LWP 25871)]
[New Thread 0xeacfcb40 (LWP 25872)]
[Thread 0xeaefeb40 (LWP 25870) exited]
[Thread 0xec3feb40 (LWP 25867) exited]
[New Thread 0xeabfbb40 (LWP 25873)]
[Thread 0xec4ffb40 (LWP 25866) exited]
[New Thread 0xec3feb40 (LWP 25874)]
[New Thread 0xd9c35b40 (LWP 25893)]
[New Thread 0xec4ffb40 (LWP 25966)]
[New Thread 0xd9434b40 (LWP 25968)]

Thread 1 "steam" received signal SIGTRAP, Trace/breakpoint trap.
0xefb2e186 in ?? () from /home/cbm80/.steam/ubuntu12_32/vgui2_s.so
(gdb) 

I no longer think the CRITICAL message mentioned above is related to this problem. I reran the same test as I did in #5294 a few more times, but did not see the CRITICAL message in any of these additional tests.

Looks good so far and everything seems to be working!

Running Steam on debian 10 64-bit
STEAM_RUNTIME is enabled automatically
Pins up-to-date!
/home/cbm80/.steam/ubuntu12_32/steam -nominidumps -nobreakpad
[2020-04-24 15:50:47] Startup - updater built Apr  4 2020 00:37:13
[2020-04-24 15:50:47] Verifying installation...
[2020-04-24 15:50:48] Verification complete
STEAM_RUNTIME_HEAVY: ./steam-runtime-heavy