Reland "[Profile creation] Split Dice-specific signin logic from the generic one"

This is a reland of b9f7f787

Original change's description:
> [Profile creation] Split Dice-specific signin logic from the generic one
>
> This CL splits the Dice-specific sign-in logic (now called
> ProfilePickerDiceSignInProvider) from the Dice-agnostic signed-in
> profile creation logic (now called ProfilePickerSignedInFlowController).
> This unblocks reusing the generic part of the code for lacros. It is
> (almost) pure refactoring, with no noticeable behavioral changes (that
> only requires minor changes in tests).
>
> To simplify reviewing, both new classes are kept in the same .h and .cc
> files. They will be split into separate units in a trivial follow-up.
>
> Bug: 1227029
> Change-Id: Ic6e0df8f7c3543c69fbd872cc52e9f57c8a0c72a
> Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3107008


> Commit-Queue: Jan Krcal <jkrcal@chromium.org>
> Reviewed-by: Stefan Kuhne <skuhne@chromium.org>
> Reviewed-by: Kush Sinha <sinhak@chromium.org>
> Cr-Commit-Position: refs/heads/main@{#915904}

Bug: 1227029
Change-Id: I1f56a341e105fef89d4e8c7720c1e9aec3eaf8a2
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3124818


Reviewed-by: David Roger <droger@chromium.org>
Commit-Queue: Jan Krcal <jkrcal@chromium.org>
Cr-Commit-Position: refs/heads/main@{#918745}