If there were fewer Chiltern services from Amersham/Chesham to
Harrow-on-the-Hill, then London Underground would have to run more trains,
which would just slow the Chiltern Trains. (I am assuming that peak hour
trains every 30 minutes would be unacceptable?)

Considering London Underground's track-record (?!) surely it would be better
to hand over this branch to Chiltern and allow them to run trains according
to their timetable, some could be all-stoppers and some express etc.