Following advices from other contributors I’m posting here the “solution” for future reference, just in case someone is experiencing the same issues.
As I wrote in one of the comments, while investigating the REST API problem – by the way, I’d have never narrowed it down without some help from @SallyCJ – I found out some malicious files in all three WordPress installs. I wasn’t looking for malware or exploits, I simply stumbled upon some suspicious php files that weren’t supposed to be on my server. Hosting support helped me doing a thorough security scan, so I could remove all the nasty files.
I don’t know why but when the malicious files where detected the support staff automatically downgraded my PHP version from 7.3 to 7.1.
Honestly, I don’t know what did the trick – the cleaning or php downgrade – and even the support staff had no clue, but right now all the websites are working, I can post new content using the block editor and everything seem to work just fine.