ExtensionDownloader uses IdentityProvider, which in practice is ProfileIdentityProvider, as the only place in the codebase where this is set is here:

https://cs.chromium.org/chromium/src/chrome/browser/extensions/updater/chrome_extension_downloader_factory.cc?type=cs&q=%22SetWebstoreIdentityProvider(%22&sq=package:chromium&l=70

This should be converted to be usage of the Identity Service client library.