I have heard that it is a bit tricky to persuade Ubuntu to work together with a newer version. But if it is needed, I would try.
Or is there an alternative for the <max_concurrent> to be used in the old BM ?

You could try

https://launchpad.net/~costamagnagianfranco/+archive/ubuntu/boinc

I use that on Linux Mint 20.3. It doesn't install on Mint 21.1, which I think is based on Ubuntu 20.4, but that may have different security policies. If you can load it, it's been reliable for me.

No, I don't want that any project will override parameters.

Then use local preferences set in each host's Boinc Manager.

I have heard that it is a bit tricky to persuade Ubuntu to work together with a newer version. But if it is needed, I would try.
Or is there an alternative for the <max_concurrent> to be used in the old BM ?

Instead of <max_concurrent> and specifying the app name you could try <project_max_concurrent> which appears to be more reliable.

He says he's running "I run Ubuntu 20.04LTS with BM 7.16.6 which is the latest one for Ubuntu20."

By BM, I take it to mean mean the v7.16.6 package of BOINC Manager+client. He's also running app_config.xml files with <max_concurrent> entries: that combination is known to be problematic.

Short of running experimental versions, the answer is probably to watch out for v7.20.5 or v7.22.x being added to the Ubuntu repositories, and upgrade as soon as possible.

Add to above:
While setting the cache to al low value may sound to be a good way to reduce the load on your computer it doesn't work that way. The cache is calculated from the average task duration and the number of CPU threads being used. So, if you have a 32 thread CPU and have your cache set to 1 day's worth of work you will get 32 CPU-days worth of work, and BOINC will try to run 32 tasks at a time. What you may find works is to limit the number of CPU threads BOINC can use, and not being a regular Linux user I won't suggest where to look for the appropriate file to edit to control this. However if you are using the GUI you can look under "options"/"computing preferences" and reduce the value of "use at most xx% of CPUs" to say 50%.

How much memory on the affected computer(s)?

Asteroids:
<app_config>
  <app>
   <name>period_search</name>
   <max_concurrent>9</max_concurrent>
  </app>
</app_config>

TNgrid:
<app_config>
  <app>
   <name>gene_pcim</name>
   <max_concurrent>20</max_concurrent>
  </app>
</app_config>

Einstein:
<app_config>
   <app>
   <name>hsgamma_FGRPB1G</name>
   <max_concurrent>2</max_concurrent>
     <gpu_versions>
         <gpu_usage>0.5</gpu_usage>
         <cpu_usage>1</cpu_usage>
     </gpu_versions>
   </app>
</app_config>