Move recents-during-split handling to MixedHandler

This is a mixed operation so it shouldn't be handled
internally to split. Also, this will be needed once
recents are directly requested vs. coming from Core.

Bug: 269691848
Test: moving code around, so existing tests pass.
Change-Id: Ia8aeaace6cb3fca04b641c6c1c1890d06f1bdbd0