crash when quickly selecting SE from Private Tab and then closing Settings via X

[kjozwiak]

Description

When you quickly select a SE from Private Tab and then close the Search engines settings page via X, the browser ends up crashing from time to time. I've reproduced the crash several times while running through brave/brave-core#10590.

Crashes:

• https://brave.sp.backtrace.io/p/brave/debug?filters=(_rxid%3Ddad20000-27c7-9009-0000-000000000000)&fingerprint=ebe50b8bb8f7604e067fdd27668bb66628bca37334238b4439ad2fcb26b34af9&debug=(%2244f79e6%22,0,0)
• https://brave.sp.backtrace.io/p/brave/debug?filters=(_rxid%3Dd85c0100-27c7-9009-0000-000000000000)&fingerprint=ebe50b8bb8f7604e067fdd27668bb66628bca37334238b4439ad2fcb26b34af9

Steps to reproduce

• install 1.49.55 Chromium: 109.0.5414.86
• launch Brave and quickly go through onboarding (default browser/P3A/Permissions on Android 13)
• go into Hamburger Menu -> Settings -> Search engines
• tap on Standard and select Bing
• tap on < to back to Settings and tap on Private
• select Qwant and then quickly tap on the X to dismiss the Search engines settings page

Brave will usually crash within a ~ms once the Search engines settings page is closed.

Actual result

screen-20230202-140340.mp4

Expected result

Brave shouldn't be crashing once a user selects a SE via Private Tab and closes the settings page via X

Issue reproduces how often

100% reproducible using the STR/Cases outlined above.

Version/Channel Information:

• Can you reproduce this issue with the current Play Store version? Yes
• Can you reproduce this issue with the current Play Store Beta version? `Yes
• Can you reproduce this issue with the current Play Store Nightly version? Yes

Device details

• Install type (ARM, x86): ARM
• Device type (Phone, Tablet, Phablet): Pixel 6 (Phone)
• Android version: Android 13

Brave version

Brave | 1.49.55 Chromium: 109.0.5414.87 (Official Build) canary (32-bit)
--- | ---
Revision | 2dc18eb511c56e012081b4abc9e38c81c885f7d4-refs/branch-heads/5414@{#1241}
OS | Android 13; Build/TQ1A.230105.002

Website problems only

• Does the issue resolve itself when disabling Brave Shields? N/A
• Does the issue resolve itself when disabling Brave Rewards? N/A
• Is the issue reproducible on the latest version of Chrome? N/A

Additional information

[kjozwiak]

The above requires 1.48.164 or higher for 1.48.x verification 👍

[Uni-verse]

Verified on Samsung GS 21 using version:

Brave	1.48.166 Chromium: 110.0.5481.100 (Official Build) (64-bit) 
Revision	4be7a36f7cb943af6118e449bbab494b43dcaddd-refs/branch-heads/5481_77@{#14}
OS	Android 13; Build/TP1A.220624.014

Using STR in #28262 (comment), ensured that crash doesn't occur when selecting SE from private tab and then quickly closing settings with X

28262_fix3.mp4