Private window's search provider don't use normal windows' when prefs is turned off

[simonhong]

Description

When alternative search engine prefs turned off, private window uses normal window's search engine provider.

When user changes provider from settings, private window should use it if this prefs turned off after private profile is created.

Steps to Reproduce

1. Launch browser
2. Open private window and turn off ddg
3. Change search engine provider from settings
4. Check private windows' search engine provider is same with normal windows'

Actual result:

Private windows' search engine provider isn't changed

Expected result:

Private windows' should use same provider with normal windows

Reproduces how often:

Brave version (chrome://version info)

Reproducible on current release:

• Does it reproduce on brave-browser dev/beta builds?
• Does it reproduce on browser-laptop?

Website problems only:

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

Additional Information

[btlechowski]

Verification passed on

Brave	0.55.12 Chromium: 70.0.3538.45 (Official Build) (64-bit)
Revision	cbdc32e4334458954e9def214d7e5fa1ca1960eb-refs/branch-heads/3538@{#830}
OS	Windows 7

Verified passed with

Brave	0.55.12 Chromium: 70.0.3538.45 (Official Build) beta(64-bit)
Revision	cbdc32e4334458954e9def214d7e5fa1ca1960eb-refs/branch-heads/3538@{#830}
OS	Mac OS X

Verification passed on

Brave	0.55.12 Chromium: 70.0.3538.45 (Official Build) (64-bit)
Revision	cbdc32e4334458954e9def214d7e5fa1ca1960eb-refs/branch-heads/3538@{#830}
OS	Linux