Move PrintBackendServiceManager static variables into class members

The PrintBackendServiceManager class has made use of static variables
for some management logic.  This has worked fine until now since there
is only ever one instance of the class created in the browser process.

This becomes a problem for expanded testing, where the results of one
test run could persist into a following run, and the residual state
could now be an issue for providing a deterministic result for following
runs.

Avoid this problem by removing these static variables and replacing
them with corresponding class members in PrintBackendServiceManager.
This ensures they get any necessary reinitialization at the start of
each run.

Bug: 1487329
Change-Id: I84c7f66a0e43466745f2b9dad4e1c8feaba1e607
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4932687
Reviewed-by: Andy Phan <andyphan@chromium.org>
Commit-Queue: Alan Screen <awscreen@chromium.org>
Code-Coverage: findit-for-me@appspot.gserviceaccount.com <findit-for-me@appspot.gserviceaccount.com>
Cr-Commit-Position: refs/heads/main@{#1209104}