Search extension is unable to change default search engine on Linux

[btlechowski]

Follow up to #10601

Search extension should be able to change default search engine, but can't.

Steps to Reproduce

1. Clean profile
2. Install ecosia extension: https://chrome.google.com/webstore/detail/ecosia-the-search-engine/eedlgdlajadkbbjoobobefphmfkcchfk/related
3. Search test in Omni bar

Test case for Private Window can be found in brave/brave-core#8626

Actual result:

google is used as default engine

Expected result:

ecosia is used as default engine

Reproduces how often:

100% repro rate

Brave version (brave://version info)

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 nightly channel? yes

Brave	1.27.39 Chromium: 91.0.4472.77 (Official Build) nightly (64-bit)
Revision	1cecd5c8a856bc2a5adda436e7b84d8d21b339b6-refs/branch-heads/4472@{#1246}
OS	Linux

cc @simonhong @rebron

[simonhong]

@btlechowski This is upstream behavior. Chrome also doesn't set extenion's search provider as a default.
Need to investigate why this is blocked on linux.