Run new PostgreSQL backend tests

[AlekSi]

Description

Closes #3126.

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 #3407 (08a7792) into main (4e6b869) will decrease coverage by 0.06%.
The diff coverage is 0.00%.

@@            Coverage Diff             @@
##             main    #3407      +/-   ##
==========================================
- Coverage   74.68%   74.63%   -0.06%     
==========================================
  Files         414      414              
  Lines       25747    25747              
==========================================
- Hits        19230    19215      -15     
- Misses       5375     5387      +12     
- Partials     1142     1145       +3     

Files Changed	Coverage
internal/backends/backend.go	ø
internal/backends/collection.go	ø
internal/backends/database.go	ø
internal/handlers/sqlite/sqlite.go	0.00%

Flag	Coverage Δ
filter-true	70.05% <0.00%> (?)
hana-1	?
hana-2	?
hana-3	?
integration	70.05% <0.00%> (-0.06%)	⬇️
mongodb-1	4.69% <0.00%> (ø)
pg-1	43.14% <0.00%> (-0.04%)	⬇️
pg-2	42.91% <0.00%> (ø)
pg-3	43.35% <0.00%> (-0.11%)	⬇️
postgresql-1	∅ <ø> (?)
postgresql-2	∅ <ø> (?)
postgresql-3	∅ <ø> (?)
pushdown-false	?
pushdown-true	?
sort-false	70.05% <0.00%> (?)
sqlite-1	42.10% <0.00%> (-0.01%)	⬇️
sqlite-2	41.56% <0.00%> (-0.01%)	⬇️
sqlite-3	42.37% <0.00%> (-0.14%)	⬇️
unit	24.73% <0.00%> (-0.02%)	⬇️

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[chilagrow]

Thank you for this!

[chilagrow]

🚀