Shard integration tests

[AlekSi]

Description

This PR improves, simplifies, and refactors sharding logic, and hooks it into CI.

Branch protection rules should be updated, hence do not merge PRs that should not be merged .

Readiness checklist

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

[codecov]

Codecov Report

Merging #2820 (1d966ad) into main (38776c6) will not change coverage.
The diff coverage is n/a.

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #2820   +/-   ##
=======================================
  Coverage   63.79%   63.79%           
=======================================
  Files         442      442           
  Lines       22523    22523           
=======================================
  Hits        14368    14368           
  Misses       7216     7216           
  Partials      939      939           

Flag	Coverage Δ
integration	56.83% <ø> (ø)
mongodb	5.00% <ø> (ø)
pg	56.74% <ø> (ø)
shard-1	43.22% <ø> (?)
shard-2	44.62% <ø> (?)
shard-3	41.38% <ø> (?)
unit	25.19% <ø> (-0.05%)	⬇️

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

[rumyantseva]

This looks awesome! Looking forward to seeing it in action!

[noisersup]

That looks great!