Move Channel::get_update_time_counter and some other methods

This is one of a series of commits to make sure methods are moved by
chunks so they are easily reviewable in diffs. Unfortunately they are
not purely move-only as fields need to be updated for things to
compile, but these should be quite clear.

This commit also uses these methods through the `context` field where
needed for compilation and tests to pass due to the above change.

Author: dunxen