Make testutil.Logger easier to use

[AlekSi]

Readiness checklist

• I added/updated unit tests.
• I added/updated integration/compatibility tests.
• I added/updated comments and checked rendering.
• I made spot refactorings.
• I updated user documentation.
• I ran task all, and it passed.
• I ensured that PR title is good enough for the changelog.
• (for maintainers only) I set Reviewers (@FerretDB/core), Labels, Project and project's Sprint fields.
• I marked all done items in this checklist.

[codecov]

Codecov Report

Merging #2790 (bd8a276) into main (200c42a) will decrease coverage by 36.77%.
The diff coverage is 87.50%.

Additional details and impacted files

@@             Coverage Diff             @@
##             main    #2790       +/-   ##
===========================================
- Coverage   62.84%   26.07%   -36.77%     
===========================================
  Files         441      441               
  Lines       22737    22740        +3     
===========================================
- Hits        14288     5929     -8359     
- Misses       7517    16201     +8684     
+ Partials      932      610      -322     

Impacted Files	Coverage Δ
integration/setup/setup_compat.go	9.28% <0.00%> (-67.15%)	⬇️
integration/setup/setup.go	62.08% <100.00%> (-1.10%)	⬇️
internal/util/testutil/testutil.go	100.00% <100.00%> (ø)

... and 142 files with indirect coverage changes

Flag	Coverage Δ
integration	4.94% <62.50%> (-51.38%)	⬇️
mongodb	4.94% <62.50%> (+<0.01%)	⬆️
pg	?
unit	24.54% <100.00%> (-0.04%)	⬇️

Flags with carried forward coverage won't be shown. Click here to find out more.

[noisersup]

LGTM

[rumyantseva]

👌