Allow usage without state dir

[AlekSi]

Description

Closes #3699.

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 #3703 (8cf88dc) into main (3b937b1) will increase coverage by 44.32%.
The diff coverage is n/a.

@@             Coverage Diff             @@
##             main    #3703       +/-   ##
===========================================
+ Coverage   31.89%   76.21%   +44.32%     
===========================================
  Files         323      323               
  Lines       23029    23029               
===========================================
+ Hits         7344    17552    +10208     
+ Misses      15012     4437    -10575     
- Partials      673     1040      +367     

see 183 files with indirect coverage changes

Flag	Coverage Δ
filter-true	72.24% <ø> (+65.90%)	⬆️
hana-1	?
integration	72.24% <ø> (+65.90%)	⬆️
mongodb-1	5.47% <ø> (ø)
postgresql-1	52.43% <ø> (∅)
postgresql-2	51.63% <ø> (∅)
postgresql-3	50.69% <ø> (∅)
sort-false	72.24% <ø> (+65.90%)	⬆️
sqlite-1	51.78% <ø> (∅)
sqlite-2	50.86% <ø> (∅)
sqlite-3	49.90% <ø> (∅)
unit	30.08% <ø> (ø)

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