Unfortunately, there was no saving the installation.
In the call with Apple, they told me that there are some corrupted files on the system that can not be repaired, and they advised fully formatting and reinstalling the system.
Apple diagnostics, NVRAM/PRAM and SMC reset and other usual stuff did not help. I did not find command line tools that could scan and fix the disk in the spirit of SFC and DISM commands on Windows.
I used external USB SSD with mac installation to access files, copied them to another drive (2x) and then bombed the system and reinstalled it.
It is a hassle, but unfortunately nothing else would have helped.
I have reinstalled the mac using USB stick with latest Ventura on it and all now seems dandy. In retrospect, Windows machines seem much more repairable than Mac when c*ap hits the fan, but it is what it is.
I presume that the issue I faced was due to relatively old system (about 5y) that had been upgraded several times. If I have stayed on Catalina I originally used for years, this probably wouldn't have happened, for a while at least.
ventura slow boot
also brings results referring to a recent system upgrade.