tests/util/test_app: Remove obsolete db_conn() fn

[Turbo87]

None of our tests strictly need a sync database connection anymore, and those that were still using one are migrated by this commit to use an async one instead. 🎉

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 89.35%. Comparing base (f74a14e) to head (c49bf69).
Report is 3 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main   #10050      +/-   ##
==========================================
- Coverage   89.40%   89.35%   -0.06%     
==========================================
  Files         295      295              
  Lines       31255    31277      +22     
==========================================
+ Hits        27944    27946       +2     
- Misses       3311     3331      +20     

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