Fix fluky test, refactor it

[AlekSi]

Readiness checklist

• I added tests for new functionality or bugfixes.
• I ran task all, and it passed.
• I added/updated comments for both exported and unexported top-level declarations (functions, types, etc).
• I checked comments rendering with task godocs.
• I ensured that the title is good enough for the changelog.
• (for maintainers only) I set Reviewers (@FerretDB/core), Assignee, Labels, Project and project's Sprint fields.
• I marked all done items in this checklist.

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Name	Status	Preview	Updated
ferret-db	✅ Ready (Inspect)	Visit Preview	Sep 27, 2022 at 11:23AM (UTC)

[codecov]

Codecov Report

Merging #1185 (145b602) into main (cdcf9b6) will increase coverage by 21.22%.
The diff coverage is n/a.

Additional details and impacted files

@@             Coverage Diff             @@
##             main    #1185       +/-   ##
===========================================
+ Coverage   44.94%   66.16%   +21.22%     
===========================================
  Files         250      250               
  Lines       12260    12260               
===========================================
+ Hits         5510     8112     +2602     
+ Misses       6042     3254     -2788     
- Partials      708      894      +186     

Impacted Files	Coverage Δ
internal/util/version/version.go	55.00% <0.00%> (-5.00%)	⬇️
internal/clientconn/conn.go	44.52% <0.00%> (+0.70%)	⬆️
integration/helpers.go	87.23% <0.00%> (+1.06%)	⬆️
internal/handlers/pg/pgdb/collections.go	50.00% <0.00%> (+2.27%)	⬆️
internal/handlers/pg/pgdb/settings.go	39.53% <0.00%> (+2.32%)	⬆️
internal/handlers/pg/pgdb/tables.go	64.70% <0.00%> (+2.94%)	⬆️
internal/handlers/pg/pgdb/query.go	40.68% <0.00%> (+4.82%)	⬆️
internal/handlers/pg/pgdb/databases.go	55.00% <0.00%> (+5.00%)	⬆️
internal/types/document.go	86.69% <0.00%> (+8.25%)	⬆️
internal/handlers/common/limit.go	75.00% <0.00%> (+8.33%)	⬆️
... and 48 more

Flag	Coverage Δ
integration	62.45% <ø> (+25.46%)	⬆️
mongodb	14.80% <ø> (-0.03%)	⬇️
pg	51.86% <ø> (?)
tigris	34.44% <ø> (-0.03%)	⬇️
unit	23.87% <ø> (-0.15%)	⬇️

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

[rumyantseva]

LGTM

[noisersup]

LGTM, I've checkouted to the branch and there are still TestFindAndModifyCompatErrors that are failing locally, but I assume that's not the scope of this PR