[New solution] Fix race condition (also a regression of the PR 19139)

[ahrtr]

Fix #19172

Original solution: #19221

cc @serathius @fuweid

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[codecov]

Codecov Report

Attention: Patch coverage is 88.88889% with 4 lines in your changes missing coverage. Please review.

Project coverage is 68.85%. Comparing base (363b166) to head (f330406).

Files with missing lines	Patch %	Lines
server/embed/etcd.go	77.77%	4 Missing ⚠️

Additional details and impacted files

Files with missing lines	Coverage Δ
server/embed/serve.go	59.38% <100.00%> (ø)
server/embed/etcd.go	76.49% <77.77%> (+0.29%)	⬆️

... and 26 files with indirect coverage changes

@@            Coverage Diff             @@
##             main   #19257      +/-   ##
==========================================
+ Coverage   68.83%   68.85%   +0.02%     
==========================================
  Files         420      420              
  Lines       35670    35673       +3     
==========================================
+ Hits        24552    24563      +11     
+ Misses       9697     9690       -7     
+ Partials     1421     1420       -1     

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[ahrtr]

cc. @fuweid @serathius PTAL

This PR is only for main branch, no backport.