Test suite: start porting to new implementation of Persistent Storage

This includes:

 - Use Dogtail, so we exercise accessibility at least a little.

 - Use the remote shell's ability to execute Python

   Let's not bother converting a string into a Python one-liner to execute it
   through a shell: nowadays, the remote shell has native Python support :)

 - Check initial state before modifying it

   If the next assertion ever fails, this one will give us confidence
   that at least we started from the state we expected.

refs #17803