SG-44442 Fix published files name search for flowam

[carlos-villavicencio-adsk]

This pull request refines the filtering logic in proxymodel_latestpublish.py to ensure that search filtering is always applied before type filtering. This change improves the accuracy and predictability of the filter behavior, especially when both search and type filters are used.

Filtering logic improvements:

• The search filter is now applied before checking for type filters, ensuring that search results are always respected regardless of the type filter state.
• If the search filter is set but the item's searchable field is missing or empty, the item is now correctly excluded from the results.
• The code clarifies that when no type filter is active (_valid_type_ids is None), all items passing the search filter are accepted.

[codecov]

Codecov Report

❌ Patch coverage is 0% with 4 lines in your changes missing coverage. Please review.
✅ Project coverage is 18.27%. Comparing base (1a00055) to head (614f38e).

Files with missing lines	Patch %	Lines
python/tk_multi_loader/proxymodel_latestpublish.py	0.00%	4 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #156      +/-   ##
==========================================
- Coverage   18.28%   18.27%   -0.02%     
==========================================
  Files          32       32              
  Lines        3074     3076       +2     
==========================================
  Hits          562      562              
- Misses       2512     2514       +2     

Flag	Coverage Δ
Linux	18.27% <0.00%> (-0.02%)	⬇️
Python-3.10	18.27% <0.00%> (-0.02%)	⬇️
Python-3.11	18.27% <0.00%> (-0.02%)	⬇️
Python-3.13	18.27% <0.00%> (-0.02%)	⬇️
Python-3.9	18.27% <0.00%> (-0.02%)	⬇️
Windows	18.27% <0.00%> (-0.02%)	⬇️
macOS	18.27% <0.00%> (-0.02%)	⬇️

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[chenm1adsk]

LGTM!