Add brave://path to support deep linking to Search Engine edit popup in settings

[bradleyrichter]

Description

In order to open the proper dialog allowing users to change their default search engine, we need to add a deep link path to the settings popup.

This change is dependent on this issue which will allow for targeting the correct popup depending on normal or private settings.

#42990

Steps to reproduce

try to use the path: brave://settings/search

notice it doesn't open the popup behind the Change button

Actual result

no path to open sub dialog

Expected result

link available to this page:

Reproduces how often

Easily reproduced

Brave version (brave://version info)

all versions

Channel information

• release (stable)
• beta
• nightly

Reproducibility

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

Miscellaneous information

No response