[9.20] chg: test: Various isctest.asyncserver refactors

Several changes to the interface of isctest.asyncserver has been made
during the development of the rewrites, propagate these to all the others.

Others have become clear as well but have been deferred as multiple
rewrites have been in review at once.

Introduce all of these in this MR.

Backport of MR !11179

Merge branch 'backport-stepan/asyncserver-improvements-9.20' into 'bind-9.20'

See merge request isc-projects/bind9!11389