WebUI: Remove pre-CR2023 code in a few WebUI surfaces, part 2.

in this part updating,
 - access_code_cast
 - downloads
 - webui_gallery
 - new_tab_page
 - new_tab_page_incognito

Bug: 326116364
Change-Id: If2d9937d39e8880cd749ed0ebc203932934bdb2b
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5478768


Commit-Queue: Demetrios Papadopoulos <dpapad@chromium.org>
Reviewed-by: John Lee <johntlee@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1291984}