Search engine favicon should be updated when a different search engine is selected via customize dashboard

[MadhaviSeelam]

Description

Found while testing https://github.com/brave/brave-browser/issues/38675. When clicked Customize list via Search widget and changed NTP search engine in the Customize Dashboard, favicon of the search engine is not updated in the Search widget. However selected search engine is shown in the dropdown as expected. This is happening in in both Windows 11 (1.67.101) & Linux (1.67.104)

Steps to reproduce

1. Installed 1.67.104
2. launched Brave
3. enabled Brave Search Widget on the NTP via brave://flags
4. open a new tab and click on Brave icon in the search widget
5. click Customize list
6. unselect Brave and select Qwant
7. close (x) the dashboard
8. open a new tab and verify the favicon in the search widget

Actual result

Brave favicon is shown. Had to select the search engine to populate the widget.

Expected result

Icon should be updated to Qwant logo

Reproduces how often

Easily reproduced

Brave version (brave://version info)

Brave 1.67.104 Chromium: 126.0.6478.26 (Official Build) beta (64-bit) Revision 9a4d1cb2f47a60953416cd3f39f0520e4dd4cf0e OS Linux

Channel information

• [ ] release (stable)
• [X] beta
• [X] nightly

Reproducibility

• [ ] with Brave Shields disabled
• [ ] with Brave Rewards disabled
• [ ] in the latest version of Chrome

Miscellaneous information

@rebron cc: @brave/qa-team

[stephendonner]

@MadhaviSeelam mind editing to include the brave://flags portion in the steps, for posterity? 🙇

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