refactor: convert HeaderMenu to functional component

[adamalston]

Closes #19126
Partially addresses #19144

Converted HeaderMenu to a functional component.

Changelog

New

• Added missing types to HeaderMenu.

Changed

• Converted HeaderMenu to a functional component.

Testing / Reviewing

yarn test packages/react

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[codecov]

Codecov Report

Attention: Patch coverage is 92.15686% with 4 lines in your changes missing coverage. Please review.

Project coverage is 84.86%. Comparing base (577e609) to head (01679e4).
Report is 25 commits behind head on main.

Files with missing lines	Patch %	Lines
...ckages/react/src/components/UIShell/HeaderMenu.tsx	92.15%	3 Missing and 1 partial ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main   #19128      +/-   ##
==========================================
- Coverage   84.87%   84.86%   -0.01%     
==========================================
  Files         372      372              
  Lines       14430    14421       -9     
  Branches     4750     4694      -56     
==========================================
- Hits        12247    12238       -9     
- Misses       2036     2037       +1     
+ Partials      147      146       -1     

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[Gururajj77]

Looks good