Don't remove background task immediately on screenshot service end

We don't want to stop the screenshot from continuing to save in
the background. The background task is only used for saving the
screenshot (and only with the shelf UI flag turned on).

Flag: NONE
Bug: 329659738
Test: manual (ensure that if screenshot service ends early, screenshot
is still saved)

Change-Id: I2c1efbf14e4c690a1e6aba7c870c0b79f437f571
Merged-In: I2c1efbf14e4c690a1e6aba7c870c0b79f437f571