Make our own low-level driver for testing

[noisersup]

Description

Closes #4146.

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

Attention: Patch coverage is 35.13514% with 24 lines in your changes are missing coverage. Please review.

Project coverage is 73.99%. Comparing base (87e7082) to head (79d4319).

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #4193      +/-   ##
==========================================
- Coverage   74.60%   73.99%   -0.61%     
==========================================
  Files         339      339              
  Lines       22876    22910      +34     
==========================================
- Hits        17067    16953     -114     
- Misses       4502     4660     +158     
+ Partials     1307     1297      -10     

Files	Coverage Δ
internal/driver/driver.go	43.80% <35.13%> (+22.68%)	⬆️

... and 17 files with indirect coverage changes

Flag	Coverage Δ
filter-true	67.39% <ø> (-0.78%)	⬇️
hana-1	?
integration	67.39% <ø> (-0.78%)	⬇️
mongodb-1	5.14% <ø> (+<0.01%)	⬆️
postgresql-1	46.57% <ø> (+0.13%)	⬆️
postgresql-2	49.54% <ø> (-0.10%)	⬇️
postgresql-3	49.84% <ø> (+0.05%)	⬆️
sqlite-1	45.69% <ø> (ø)
sqlite-2	48.74% <ø> (-0.08%)	⬇️
sqlite-3	48.97% <ø> (-0.03%)	⬇️
unit	32.69% <35.13%> (+0.16%)	⬆️

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

[noisersup]

@AlekSi some questions to you, thanks!

[chilagrow]

Looking good overall, see the linter warnings