Cleanup awaitData tailable cursor

[noisersup]

Description

Closes #2283.

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

All modified and coverable lines are covered by tests ✅

Comparison is base (f8d94bd) 74.87% compared to head (65a3e2a) 73.87%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #3915      +/-   ##
==========================================
- Coverage   74.87%   73.87%   -1.00%     
==========================================
  Files         330      330              
  Lines       25665    25671       +6     
==========================================
- Hits        19217    18965     -252     
- Misses       5280     5577     +297     
+ Partials     1168     1129      -39     

Files	Coverage Δ
internal/handler/msg_getmore.go	84.27% <100.00%> (+0.38%)	⬆️

... and 15 files with indirect coverage changes

Flag	Coverage Δ
filter-true	70.56% <100.00%> (-1.31%)	⬇️
hana-1	?
integration	70.56% <100.00%> (-1.31%)	⬇️
mongodb-1	5.22% <0.00%> (-0.01%)	⬇️
postgresql-1	50.77% <100.00%> (+0.02%)	⬆️
postgresql-2	52.19% <100.00%> (+0.03%)	⬆️
postgresql-3	50.29% <92.30%> (+0.11%)	⬆️
sqlite-1	49.92% <100.00%> (-0.08%)	⬇️
sqlite-2	51.27% <100.00%> (+0.07%)	⬆️
sqlite-3	49.41% <92.30%> (+0.12%)	⬆️
unit	30.74% <0.00%> (-0.01%)	⬇️

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[rumyantseva]

LGTM