Navigation bar styling for Downloads manager not reflected in Private or Tor Window mode

[cndouglas]

Description

The navigation bar at the top of the Downloads page (that shows Settings, Brave Rewards, History, Bookmarks, and Downloads) looks different when the Downloads page is viewed in a private window. This only applies to the Downloads page because the other pages may only be opened in a normal window.

Steps to Reproduce

1. Open a new private window.
2. Go to brave://downloads/.

Actual result:

The navigation bar has a solid blue background and uses a serif font.

Expected result:

The navigation bar should match the normal one with a pink/purple background and a sans-serif font.

Reproduces how often:

Always

Brave version (brave://version info)

Brave | 0.63.9 Chromium: 73.0.3683.75 (Official Build) dev(64-bit)
Revision | 909ee014fcea6828f9a610e6716145bc0b3ebf4a-refs/branch-heads/3683@{#803}
OS | Mac OS X

Reproducible on current release:

• Does it reproduce on brave-browser dev/beta builds? Yes, 0.63.9.

Website problems only:

• Does the issue resolve itself when disabling Brave Shields? Not applicable.
• Is the issue reproducible on the latest version of Chrome? Not applicable.

Additional Information

[bsclifton]

cc: @petemill

[rebron]

Related to #3503.
This issue is showing up on Guest Windows and Tor Window (because it's set up a guest window). Private window is working properly as its showing settings in a normal window.

[rebron]

Still an issue with Downloads page. The other pages like settings and extensions are being redirected to normal window mode.