I can't know that BR would have made improvements. No-one can know what would have happened. But I do know that the private sector bits of the railway haven't made any improvements, and I'm unapologetic about questioning what privatisation has delivered for me.
You do realise the timetable (frequency and journey time) is dictated by DfT in the franchise ITT?
Also heavily constrained by what Network Rail can squeeze out of the infrastructure, and by PTE demands in their areas of interest.
For years, the issue on the route has been how many trains per hour can be run Manchester-Huddersfield-Leeds, and the stopping pattern.
Today's answer is 6tph TPE, with some trains stopping at the smaller stations (previously separate Northern stoppers).
It's not Northern/TPE's fault that the timetable pattern is as it is now.
BR started the "frequent, short" train pattern during its Sprinterisation phase.
I don't think you can say what BR might have done with the route had it still been in charge, but they wouldn't have had a free hand.
You can complain at TPE and Northern over on-board standards, day to day operation and some fares (not regulated ones including seasons which are DfT/PTE).
Pacers, and short 185s, are still around thanks to DfT policy of not allowing new trains in the last franchise, plus the heavy subsidy at Northern and TPE.
They will be gone in 18 months, as will the short 185s on most services, thanks to private sector funding of new trains.