Improve build + dist install checks

[Kami]

Description

This is an attempt to improve existing build + dist + install checks in an attempt to automatically catch issues such as the one reported in #2047.

Keep in mind though that it's not yet fully clear to me why the existing checks didn't catch that issue (possibly something related to the execution environment there).

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 83.40%. Comparing base (42bf483) to head (abda09e).
⚠️ Report is 56 commits behind head on trunk.

Additional details and impacted files

@@           Coverage Diff           @@
##            trunk    #2054   +/-   ##
=======================================
  Coverage   83.40%   83.40%           
=======================================
  Files         353      353           
  Lines       81684    81684           
  Branches     8631     8631           
=======================================
  Hits        68124    68124           
  Misses      10738    10738           
  Partials     2822     2822           

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