Move most DefaultMixedHandler logic to MixedTransition.

2/4 to split the the Recents functionality from the rest of
DefaultMixedHandler logic.

Bug: 314971702
Flag: NA
Test: https://android-build.corp.google.com/builds/abtd/run/L37200030000843387
Change-Id: Id285895bca02cc438bd6413ba78412c7088e5dd8