For what it's worth, I am having very similar issues since loading Catalina. MacBook Pro Late 2013 (16gb RAM, 500gb hard drive, etc. etc.). Crashes mainly when I am away from the Mac - possibly instead of sleeping, although I have even set the screensaver to never and fiddled with the power settings. However, more worryingly, it has crashed whilst using.
I have brought this to the attention of Apple Support (who, apparently, have only had one report of a similar sounding situation!!!). They have had me uploading a fresh Catalina, erasing the disc and uploading a clean new Catalina (twice), setting up a partition, deleting all 3rd-party software, gone through my logs, library, stopped iCloud Drive, etc. etc. - all over the last 2 weeks with no success. They are saying I have a hardware problem NOT a software problem.
Whilst I rate Apple's support, I think there is a breakdown in comms between them and the mothership. I suspect they were a bit adventurous in saying "late 2023 Macs" were ok. There is clearly an issue. I just want someone to be honest and not try to deny they were wrong. What was the downside - we would just go out and buy something sparkly and Apple.
I have tried downgrading to Mojave but the system crashed in the process - not sure if that was a lingering issue or what.
Like most of us, I am heavily invested in the Apple Legacy - wife, kids, iPhones, iPads, iTunes, Apple Music, etc. etc. They know we are unlikely to walk away.