[send-tab-to-self] Remove ReceivingUiHandlerRegistry

No intentional behavior change.
Before: Foo -> ReceivingUiHandlerRegistry (singleton)
        -> ReceivingUiHandler (per profile)
After: Foo -> SendTabToSelfClientService (per profile)
       -> ReceivingUiHandler (per profile)

This change also has other benefits:
- No more "virtual for testing" methods in SendTabToSelfClientService.
- No more Profile reference in SendTabToSelfClientService (see
  crbug.com/368297674).

One change worth highlighting is that in [1], only the
ReceivingUiHandler for the same profile is called now, which should be
fine.

[1] https://source.chromium.org/chromium/chromium/src/+/main:chrome/browser/send_tab_to_self/send_tab_to_self_client_service.cc;l=76;drc=079e6e95e7898ea44fb118b615eb05be2d24f8f1

Bug: 368297674
Change-Id: I3780db935bf09608b1328b8825f1ee5d99fabd1a
Fixed: 40632832
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6415804
Reviewed-by: Caroline Rising <corising@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1440581}