Start saving screenshot in background earlier

Currently, we only start saving the screenshot after the animation
into the corner is complete. Since it takes non-negligible time to
save (~.5 seconds, more if smart screenshots don't exit earlier),
and we don't pop up the actions shade until we've finished saving,
this difference is user-visible. Beginning the async saving task
earlier mitigates the apparent delay.

Bug: 148605860
Test: manual

Change-Id: Icc98fc4f9920fb89cb08654c2ab8ee6f6d3321fa