changing default SE doesn't change SE in Guest window

[LaurenWags]

Description

Found while testing #6187

When the default SE is changed, it doesn't change the SE on a Guest window. Google is still the default. However, on the tab there is a toggle to use DDG (since Guest windows use Private theme).

Not sure if this is related to #5065 - the settings page on a Guest window is empty.

Steps to Reproduce

1. Launch with a clean profile
2. Change default SE.
3. Open a Guest window.

Actual result:

Google is the default SE.

Expected result:

Default SE to be what you had chosen in settings.

Reproduces how often:

easily

Brave version (brave://version info)

Brave	0.69.135 Chromium: 77.0.3865.120 (Official Build) (64-bit)
Revision	416d6d8013e9adb6dd33b0c12e7614ff403d1a94-refs/branch-heads/3865@{#884}
OS	macOS Version 10.13.6 (Build 17G5019)

Brave	0.70.115 Chromium: 77.0.3865.90 (Official Build) beta (64-bit)
Revision	58c425ba843df2918d9d4b409331972646c393dd-refs/branch-heads/3865@{#830}
OS	macOS Version 10.13.6 (Build 17G5019)

Brave	0.72.99 Chromium: 78.0.3904.50 (Official Build) nightly (64-bit)
Revision	2accdc52c79976e264cd2694df6db31d1fccd8e8-refs/branch-heads/3904@{#658}
OS	macOS Version 10.13.6 (Build 17G5019)

Version/Channel Information:

• Can you reproduce this issue with the current release? yes
• Can you reproduce this issue with the beta channel? yes
• Can you reproduce this issue with the dev channel? yes
• Can you reproduce this issue with the nightly channel? yes

Other Additional Information:

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

Miscellaneous Information:

[btlechowski]

Reproduced on Ubuntu

Brave	0.70.115 Chromium: 77.0.3865.90 (Official Build) beta (64-bit)
Revision	58c425ba843df2918d9d4b409331972646c393dd-refs/branch-heads/3865@{#830}
OS	Ubuntu 18.04 LTS

[GeetaSarvadnya]

Reproduced in Windows 10 x64 - 0.70.116

[rebron]

Closing. Working as expected. Should use default search engine not one chosen by user in main profile.