Clean-up old code for multiple handlers

[AlekSi]

Description

Plus other small cleanups.

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 #3763 (1024fff) into main (364907c) will decrease coverage by 0.09%.
The diff coverage is 93.66%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #3763      +/-   ##
==========================================
- Coverage   76.53%   76.44%   -0.09%     
==========================================
  Files         324      324              
  Lines       23090    23090              
==========================================
- Hits        17671    17652      -19     
- Misses       4384     4402      +18     
- Partials     1035     1036       +1     

Files	Coverage Δ
...rnal/handler/common/aggregations/operators/expr.go	76.81% <ø> (ø)
.../common/aggregations/operators/operators_errors.go	100.00% <ø> (ø)
...ernal/handler/common/aggregations/operators/sum.go	95.94% <ø> (ø)
internal/handler/common/client_metadata.go	100.00% <ø> (ø)
internal/handler/commonerrors/error.go	88.23% <ø> (ø)
internal/handler/registry/registry.go	43.47% <ø> (ø)
internal/types/document.go	96.07% <ø> (ø)
internal/handler/commonparams/commonparams.go	93.66% <93.66%> (ø)

... and 8 files with indirect coverage changes

Flag	Coverage Δ
filter-true	72.49% <93.66%> (-0.09%)	⬇️
hana-1	?
integration	72.49% <93.66%> (-0.09%)	⬇️
mongodb-1	5.48% <0.00%> (ø)
postgresql-1	51.80% <64.08%> (-0.08%)	⬇️
postgresql-2	51.45% <66.90%> (ø)
postgresql-3	52.63% <82.39%> (-0.04%)	⬇️
sqlite-1	50.92% <64.08%> (ø)
sqlite-2	50.61% <66.90%> (ø)
sqlite-3	51.80% <82.39%> (+0.17%)	⬆️
unit	30.26% <45.07%> (-0.05%)	⬇️

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

[rumyantseva]

Looks good!