Tweak MongoDB initialization process

[AlekSi]

Description

And add TODOs.

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 #3307 (c965516) into main (1262111) will decrease coverage by 0.06%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##             main    #3307      +/-   ##
==========================================
- Coverage   75.22%   75.16%   -0.06%     
==========================================
  Files         400      400              
  Lines       23374    23374              
==========================================
- Hits        17582    17569      -13     
- Misses       4765     4773       +8     
- Partials     1027     1032       +5     

Files Changed	Coverage
internal/util/ctxutil/ctxutil.go	100.00%

Flag	Coverage Δ
hana	?
integration	71.76% <100.00%> (-0.05%)	⬇️
mongodb	5.05% <0.00%> (+<0.01%)	⬆️
pg	61.96% <100.00%> (+<0.01%)	⬆️
shard-1	56.58% <100.00%> (ø)
shard-2	55.60% <100.00%> (-0.05%)	⬇️
shard-3	54.34% <100.00%> (ø)
sqlite	55.39% <0.00%> (-0.06%)	⬇️
unit	24.77% <100.00%> (-0.07%)	⬇️

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

[chilagrow]

LGTM!