<<Surely registration would be more useflul than flight numbers.>>
jofm5 Yes, of course it would and many years ago commercial aircraft did use registration letters as callsign, as do the vast majority of light aircraft now. How to use the callsign is clearly explained in the R/T Manual.
Airlines use flight numbers because the flight plans for scheduled flights are stored in ATC computers worldwide. The flight plans are automatically activated for ATC purposes sometime prior to the scheduled time and all controllers who will be responsible for the flight are provided with flight plan details automatically. If registration letters were used, every one of those flight plans would either have to be individually filed or the airlines would have to notify ATC of the registrations of the aircraft on particular flights. To be honest, airlines often have trouble amending the aircraft type, which is why ATC requires pilots to confirm the type, so it woulod be a monumental nightmare messing with registrations.
Lastly, in the event of confusion ATC can request a pilot to use any callsign they choose but it doesn't happen too often.