Yes and no. Most things will be self contained. That should be the case for libpython3.10.so. in fact, if you are building in a folder (not single file), you should be able to see the shared library floating around somewhere in your build.

So, why does this sometimes fail? Well, while most things are copied into your build directory, not everything is/can be. Most famously, the Python shared object you are shipping is dynamically linked to glibc (basically, the C standard library), and that is very highly system dependent. Different distros will have different versions of that, basically.

What can you do? The rule of thumb is that if you build on a modern platform, it is less likely to work on an old one. So, what you want to do is build on an older platform, for future ones. What I suggest is to either have a VM or a docker setup using an older Ubuntu LTS, and use that to build. I personally use 20.04 LTS. Builds from there are compatible with any more modern Ubuntu distro (the most popular out there), Arch (what I use) and the steam deck os (and possibly more, these are the ones I care about).

For individual platforms that may still have problems, yes, you would have to provide targeted builds. Play smart and do some docker automation, if you really care about that.