When building Python 3.10.7 from source on Linux (I have tried this using Ubuntu focal and CentOS 8 docker images), I do something like this to create a tarball distribution:
./configure --prefix=/opt/python3
make
make install DESTDIR=install
tar -C install -cJf python3.10.tar.xz opt/python3
When I extract that tarball to / the shebang for pip3, etc. is #!/opt/python3/bin/python, but that link doesn't exist.
Either make install needs to install the python symlink or the shebang should be python3 and not python.