Guyyys... I think I just messed up. It appears I was actually using an older version of Jogamp.
I'm now using this: https://jogamp.org/deployment/archive/rc/v2.4.0-rc-20210111/fat/jogamp-fat.jar
...and it runs fine on all the JDKs.
firstname.lastname@example.org helped me out to notice the problem. If you unpack the jar and run this:
for i in $(find . -name \*.so); do echo $i; strings $i | grep SUNWprivate ; done
It should show no references to SUNWprivate. If it does, you have a very old library on your hands and should just update.
Hey, even geniuses mess up sometimes, right... ^^
So, from my POV, everything seems solved. Jogamp is ready to use :)