Update Private window search setting UX

[rebron]

Description

Selecting a default search engine for Private window should match the UX for selecting a default search engine in a normal window. For normal window, the UX was updated in 1.71.x.

Steps to reproduce

1. Go to brave://settings/search
2. Notice the difference between selecting default search engine for a normal window (button) vs private window (drop down menu)

Actual result

Expected result

Reproduces how often

Easily reproduced

Brave version (brave://version info)

1.71.x and higher

Channel information

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

Reproducibility

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

Miscellaneous information

No response

[rebron]

cc: @fmarier @rmcfadden3

[rmcfadden3]

@rebron are you just looking for me to write a string that describes the setting of a default specifically for private windows? If yes, how about:

The search engine you choose will be used for features like searching from the address bar and from images on web pages when you're using a private window.