Improve validate diagnostic command

[b1ron]

Description

Closes #419.

Readiness checklist

• I added/updated unit tests (and they pass).
• I added/updated integration/compatibility tests (and they pass).
• 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), Milestone (Next), Labels, Project and project's Sprint fields.
• I marked all done items in this checklist.

[codecov]

Codecov Report

Merging #3804 (233f9e2) into main (f4c289f) will decrease coverage by 0.14%.
The diff coverage is 100.00%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #3804      +/-   ##
==========================================
- Coverage   76.38%   76.24%   -0.14%     
==========================================
  Files         315      315              
  Lines       23540    23540              
==========================================
- Hits        17980    17949      -31     
- Misses       4519     4549      +30     
- Partials     1041     1042       +1     

Files	Coverage Δ
internal/handler/msg_validate.go	69.23% <100.00%> (ø)

... and 10 files with indirect coverage changes

Flag	Coverage Δ
filter-true	72.39% <100.00%> (-0.14%)	⬇️
hana-1	?
integration	72.39% <100.00%> (-0.14%)	⬇️
mongodb-1	5.19% <0.00%> (-0.16%)	⬇️
postgresql-1	52.34% <100.00%> (ø)
postgresql-2	51.28% <100.00%> (-0.01%)	⬇️
postgresql-3	52.98% <100.00%> (-0.03%)	⬇️
sqlite-1	51.47% <100.00%> (+0.09%)	⬆️
sqlite-2	50.43% <100.00%> (ø)
sqlite-3	52.10% <100.00%> (-0.13%)	⬇️
unit	30.73% <0.00%> (ø)

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[b1ron]

Tests are failing let me investigate.

[AlekSi]