I have split this into a new topic for you.
You would upload these to your server, in the location your site is installed. This is something that you should know as a self hosted customer there. If you are unsure, you would need to contact your hosting company.
Please could you provide more information on the incorrect cache management you are referring to? There is no information on this.
This would actually indicate caching on your server (usually opcache or varnish) which is showing incorrect files. There is no way programatically for the software to know this.
A 501 error is one thrown by the server rather than the software. Usually as a result of resource issues, but can be from other issues. Given it restarts correctly, that would indicate resource issues.
This is a plain vanilla web site, the only plug-in is a language file - en even that update goes wrong with a 501. After which I hope it is installed correctly, but not really sure.
When you say this is running on a synology NAS, this would indicate it is not being run at a hosting company as such, is this correct? Where is this being run?
I can completely understand the frustration, however from the above, these unfortuanetly do point toward hosting issues. Especially if you are experiencing this on every update. We have thousands of clients update constantly without issue.