Normally, with Linux and BOINC, just starting the Manager causes the client to start. It sounds like it is running correctly from your Start Menu selection. Does Mint have .desktop files? I would think so since is it Debian under the covers just like Ubuntu. How about creating a .desktop file for Boinc Manager. That is one of the things I had to do when I transitioned from 16.04 to 18.04, I found that in 18.04 the BOINC desktop file was not created. So I made one so that I can Dock the Manager for easy access. I don't have it autostart. This is my Boinc Manager desktop file. Don't know if this might help you. If you have a desktop file, it is just like any other desktop file you have in /usr/share/applications and can be simply added to Startup Applications. You will have to adjust directory locations for your apps.

[Desktop Entry]
Encoding=UTF-8
Version=1.0
Type=Application
Name=BOINC Manager
Icon=/home/keith/Desktop/BOINC/boincmgr.32x32.png
Path=/home/keith/Desktop/BOINC
Exec=/home/keith/Desktop/BOINC/boincmgr
StartupNotify=false
StartupWMClass=Boincmgr
OnlyShowIn=Unity;
X-UnityGenerated=true

Now I wonder, if a desktop file is the same as the desklets available for Mint? Sorry, perhaps more a question for Less Bayliss.
Anyway I'll try and try...

Normally, when Boinc is installed, the service starts at boot automatically.

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As far as the PPA sources, what's wrong with them?
Boinc is a more advanced version, and shows more projects listed.

As far as the PPA sources, what's wrong with them?

The newest versions (still in the Beta test stage - more changes are expected) have been modified by the Linux community in a way that makes them more suitable to institutional security environments. The changes haven't been universally welcomed by private individuals more used to having total control over their own computers.

Boinc is a more advanced version, and shows more projects listed.

The list of projects available is dynamic data, not tied to any particular version of BOINC. 'all_projects_list.xml' is derived from the same sources as https://boinc.berkeley.edu/projects.php.

So far nothing works and I'll give up now.
Boinc should run silently in the background and not become a major task for me (there are more important others waiting). Anyway if boinc et al want that users run this application and that it starts on a regular base i.e. automatically starts, then they should develop a working application. Just to stab around in the dark to find a solution (or actually not finding it) is just too frustrating.
Cheers.

PS: Comments like: "Normally, when Boinc is installed, the service starts at boot automatically." are not helpful, when the threadstarter described that it's not working in his system.

So far nothing works and I'll give up now.
Boinc should run silently in the background and not become a major task for me (there are more important others waiting). Anyway if boinc et al want that users run this application and that it starts on a regular base i.e. automatically starts, then they should develop a working application. Just to stab around in the dark to find a solution (or actually not finding it) is just too frustrating.
Cheers.

PS: Comments like: "Normally, when Boinc is installed, the service starts at boot automatically." are not helpful, when the threadstarter described that it's not working in his system.

That means that the threadstarter either has done something he shouldn't, or installed it in a way that it shouldn't be installed. Because otherwise, like mentioned, it should work automatically.
So trace back where you've differed from the 100s of people who successfully installed Boinc on their Linux versions?

First, I'd try installing boinc from a fresh installation of Linux.
If it doesn't work, try a generic linux, like Ubuntu or Debian, which have pretty solid support.
If it still doesn't work, chances are your hardware (or hardware combination) is to blame.

This all, assuming you've tried to manually start boinc and was successful in doing so.