Allow users to remove Leo icon from Quick Search Engines menu (and anywhere else)

[Brave-Matt]

Description

We receive regular requests/complaints from our users who do not wish to use/interact/see Leo/AI related components in the browser. While it makes sense that we would want to keep Leo visible and exposed as much as is reasonable, many of our users have come to expect that Brave will put them first and allow them to browse however they see fit.

For Android, there is no way for a user to fully "disappear" Leo in the browser at this time. Even with all Leo options toggled off, it still appears in the Quick Search Engines section, with no way to remove it:

I believe an easy solution for this would be to simply include an additional toggle, either in the Quick Search Engines menu or in Settings --> Leo that allows users to toggle off the Leo option and remove it from the quick search engines bar.

Steps to reproduce

1. Disable Leo everywhere
2. Observe quick search engines bar

Actual result

Leo icon is still there

Expected result

Users should be allowed to fully disable and/or hide Leo from the browser if desired.

Reproduces how often

Easily reproduced

Brave version

v1.76.74

Device

• Brand/model: All
• Android version: All

Channel information

• release (stable)
• beta
• nightly

Reproducibility

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

Miscellaneous information

This came in today:
https://community.brave.com/t/let-me-remove-leo-ai-button-specifically-from-quick-search/605542

But there are a litany of users on all platforms asking for this or similar functionality.

[timchilds]

@deeppandya the Leo settings should live under Settings --> Leo.

Show in address bar
Enables a shortcut to launch Leo from the address bar.

Show in quick search engines bar
Enables a shortcut to launch Leo from the quick search engines bar.

Slack discussion: https://bravesoftware.slack.com/archives/C0816QH3R/p1742319666963519?thread_ts=1742315476.024689&cid=C0816QH3R

[srirambv]

Verification passed on Google Pixel 8 with Android 15 running

Brave	1.78.72 Chromium: 135.0.7049.41 (Official Build) beta (64-bit)
Revision	bb4f6c2586a97768411682ab12bfb2f1a231d3b5
OS	Android 15; Build/BP1A.250305.019; 35; REL

• Verified steps from issue description
• Verified new setting Show in quick search engines bar is available under Settings -> Leo
• Verified setting is enabled by default
• Verified when setting is disabled, Leo is not shown in quick search engine bar

44755.mp4