[Extensions] Update developerPrivate API to use base-64 data urls for icons
Change the developerPrivate API to uniformly use base-64 data urls for icons,
rather than relying on the custom url handler.
BUG=
461039
Committed:
https://crrev.com/59f6756e6aa719a7d727ecb98456d185d1b57752
Cr-Commit-Position: refs/heads/master@{#327095}