🍒 PM-36475: Update when search icon is shown

[david-livefront]

🍒 Cherry-pick from this original PR.

🎟️ Tracking

PM-36475

📔 Objective

This PR updates when the search icon is displayed on the Vault Item Listing Screen. We were hiding the icon when not in the content state but we need to display it when in various Autofill states as ell to allow users to search for pre-existing unmatched items.

[codecov]

Codecov Report

❌ Patch coverage is 75.00000% with 1 line in your changes missing coverage. Please review.
✅ Project coverage is 85.50%. Comparing base (eace312) to head (c10ef5b).

Files with missing lines	Patch %	Lines
...t/feature/itemlisting/VaultItemListingViewModel.kt	75.00%	0 Missing and 1 partial ⚠️

Additional details and impacted files

@@                        Coverage Diff                        @@
##           release/hotfix-v2026.4.1-bwpm    #6907      +/-   ##
=================================================================
+ Coverage                          85.47%   85.50%   +0.03%     
=================================================================
  Files                                958      962       +4     
  Lines                              60965    60990      +25     
  Branches                            8646     8653       +7     
=================================================================
+ Hits                               52107    52152      +45     
+ Misses                              5857     5829      -28     
- Partials                            3001     3009       +8     

Flag	Coverage Δ
app-data	17.73% <0.00%> (+0.10%)	⬆️
app-ui-auth-tools	20.71% <0.00%> (-0.01%)	⬇️
app-ui-platform	15.92% <0.00%> (-0.01%)	⬇️
app-ui-vault	26.62% <75.00%> (+<0.01%)	⬆️
authenticator	6.56% <0.00%> (+0.02%)	⬆️
lib-core-network-bridge	4.28% <0.00%> (+<0.01%)	⬆️
lib-data-ui	1.03% <0.00%> (-0.01%)	⬇️

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[david-livefront]

Thanks @SaintPatrck