Just after writing this I decided to do another workaround and try to update again. And you know what? I succeeded! No 503 errors, no server time out, no errors and sudden stops while upgrading.
What I made different this time?
In Advanced Parameters I checked "force recompile" and "cache off". I had some warnings but I'm checking everything and the whole thing seems fine.
Of course I had the "modules not showing issue". Luckily this has an easy solution for me.
Just copy your cacert.pem archive from: public_html/*yourshop/vendor/composer/ca-bundle/res/cacert.pem
to the folder: public_html/*yourshop/app/cache/prod/cacert.pem
For safety, download this archive in a safe place and do the operation from Filezilla or your hosting backoffice.
The file must be in both folders. Just F5 after that and everything should be fine again.
Note: Probably guessed but just to make it sure, you have to change *yourshop to the folder name you have for your own very shop!
You'll get this modules weirdness everytime you modify your theme.