Simplify documents fetching for delete

[rumyantseva]

Description

Closes #898.

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.

[codecov]

Codecov Report

Merging #1584 (6e3147e) into main (babb750) will increase coverage by 0.07%.
The diff coverage is 88.00%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1584      +/-   ##
==========================================
+ Coverage   67.53%   67.60%   +0.07%     
==========================================
  Files         284      284              
  Lines       13782    13804      +22     
==========================================
+ Hits         9307     9332      +25     
+ Misses       3519     3511       -8     
- Partials      956      961       +5     

Impacted Files	Coverage Δ
internal/handlers/pg/msg_delete.go	70.31% <88.00%> (+4.43%)	⬆️
internal/handlers/pg/msg_create.go	77.21% <0.00%> (-6.33%)	⬇️
internal/util/version/version.go	48.52% <0.00%> (-4.42%)	⬇️
internal/handlers/pg/pgdb/collections.go	52.27% <0.00%> (-2.28%)	⬇️
internal/handlers/pg/pgdb/settings.go	44.21% <0.00%> (-2.05%)	⬇️
internal/handlers/common/filter.go	84.56% <0.00%> (ø)
internal/handlers/tigris/tigrisdb/collections.go	66.66% <0.00%> (+4.76%)	⬆️
internal/handlers/pg/msg_listdatabases.go	73.07% <0.00%> (+5.76%)	⬆️
internal/handlers/tigris/msg_create.go	74.64% <0.00%> (+7.04%)	⬆️
... and 1 more

Flag	Coverage Δ
integration	63.53% <88.00%> (+0.08%)	⬆️
mongodb	13.91% <0.00%> (-0.03%)	⬇️
pg	52.28% <88.00%> (+0.02%)	⬆️
tigris	36.74% <0.00%> (-0.01%)	⬇️
unit	26.78% <0.00%> (-0.07%)	⬇️

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