Cast is not disabled on toggling off media router switch in extension settings

Description

Toggling off media router setting in brave://settings/extensions should disable

Test Plan

Specified here: https://github.com/brave/brave-core/pull/10116

Steps to Reproduce

1. Enable cast by enabling the media router toggle in brave://settings/extensions
2. Verify that cast functionality works
3. Disable cast
4. Cast should not be available in the hamburger menu

Actual result:

Cast still works

Expected result:

Cast should be disabled

Reproduces how often:

Easily reproduced

Desktop Brave version:

1.29.80 Chromium: 93.0.4577.63 (Official Build) (x86_64)

[jumde]

@vis2x - The PR should be merged early next week.

[stephendonner]

Verified PASSED using

Brave	1.31.81 Chromium: 95.0.4638.40 (Official Build) beta (x86_64)
Revision	e3e7c76ba0284b16087cf4cf3153abfaef6470c7-refs/branch-heads/4638@{#624}
OS	macOS Version 11.6 (Build 20G165)

Steps:

1. enabled network filter for Little Snitch
2. new profile
3. launched Brave
4. loaded brave://settings/extensions and verifed that cast’s Media Router is disabled by default
5. verified that SSDP requests are not sent by default
6. toggled Media Router to On
7. confirmed Relaunch button is visible
8. loaded brave://flags and navigated back to brave://settings/extensions. Confirmed the Relaunch button is still visible
9. restart browser
10. verify cast works
11. toggled Media Router in brave://settings/extensions to Off
12. clicked on Relaunch
13. verified I no longer see any cast / media-router options in YouTube or the top-level browser menus

example	example	example	example	example	example

---

Verification PASSED on Samsung S10+ running Android 11 using the following build:

1.31.86 Chromium: 95.0.4638.50

• ensured that Media Router was enabled by default by visiting https://gem.cbc.ca/live/channel/ottawa and ensuring that the Cast icon was being displayed via the media player
• ensured that tapping on the Cast button displayed a modal with all the devices that can cast

Example	Example	Example	Example