Make handler accept constructed backend

[rumyantseva]

Description

Closes #3662.

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 #3710 (fb857bc) into main (6c5c709) will decrease coverage by 0.17%.
The diff coverage is 50.81%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #3710      +/-   ##
==========================================
- Coverage   76.33%   76.16%   -0.17%     
==========================================
  Files         323      323              
  Lines       23034    23043       +9     
==========================================
- Hits        17582    17550      -32     
- Misses       4412     4450      +38     
- Partials     1040     1043       +3     

Files	Coverage Δ
ferretdb/ferretdb.go	83.16% <100.00%> (+0.51%)	⬆️
integration/setup/listener.go	74.11% <100.00%> (-2.67%)	⬇️
internal/handlers/sqlite/sqlite.go	68.42% <100.00%> (-13.81%)	⬇️
internal/handlers/registry/registry.go	43.47% <33.33%> (ø)
internal/handlers/registry/postgresql.go	77.77% <62.50%> (-22.23%)	⬇️
internal/handlers/registry/sqlite.go	77.77% <62.50%> (-22.23%)	⬇️
internal/handlers/registry/hana.go	3.44% <0.00%> (-96.56%)	⬇️

... and 4 files with indirect coverage changes

Flag	Coverage Δ
filter-true	72.19% <40.98%> (-0.19%)	⬇️
hana-1	?
integration	72.19% <40.98%> (-0.19%)	⬇️
mongodb-1	5.46% <0.00%> (-0.01%)	⬇️
postgresql-1	52.31% <24.59%> (-0.06%)	⬇️
postgresql-2	51.59% <24.59%> (-0.07%)	⬇️
postgresql-3	50.67% <24.59%> (-0.05%)	⬇️
sort-false	72.19% <40.98%> (-0.19%)	⬇️
sqlite-1	51.58% <24.59%> (-0.05%)	⬇️
sqlite-2	50.84% <24.59%> (-0.01%)	⬇️
sqlite-3	49.84% <24.59%> (-0.05%)	⬇️
unit	30.09% <31.03%> (-0.02%)	⬇️

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[noisersup]

Looks great!