[lldb/test] Make it possible to run the mock gdb server on a single thread

This is a preparatory commit to enable mocking of qemu startup. That
will involve running the mock server in a separate process, so there's
no need for multithreading.

Initialization is moved from the start function into the constructor
(which can then take an actual socket instead of a class), and the run
method is made public.

Depends on D114156.

Differential Revision: https://reviews.llvm.org/D114157