Make auto enrollment check type determination logic easier to follow

Mostly stop using member variables to signal flags across multiple
function call layers.

Bug: none
Change-Id: I918b9f102012e4fdd7f5e5b5c08b63d07ea85ebe
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3086684


Commit-Queue: Denis Kuznetsov [CET] <antrim@chromium.org>
Reviewed-by: Denis Kuznetsov [CET] <antrim@chromium.org>
Reviewed-by: Roland Bock <rbock@google.com>
Cr-Commit-Position: refs/heads/main@{#923344}