The ones between Birmingham-Leicester (very few remain),
I'm one of those! When I make use of the service, that is.
It would be difficult to speed up the Birmingham-Stansted trains, as the route isn't built to support higher speeds, so missing out station stops would only save you a few minutes overall over the course of the journey, as the "fast" service would still be sibject to the same speed restrictions along the route.
Giving the service to Crosscountry has been a way to reduce the number of operators running through New Street, compared to if EMT had been allowed to run it. It wouldn't fit into the London Midland remit, seeing as they are principally bounded by the WCML and west of that, so there wasn't a great deal of choice when the Central franchise was reshaped. Plus it gives Tyseley a large fleet of 170s for maintenance, keeping things standard.
I would suggest, with a couple of others above, that the service could be split to run Birmingham-Leicester-Lincoln run by Crosscountry, and Leicester-Peterborough-Cambridge-Stansted operated by EMT. It would then be possible to concentrate three car 170s on the Crosscountry section where there are no limiting platform lengths, although I don't know how many units this would take: It would work better if Crosscountry receive the centre cars from the LM 170/6s as has been intended when LM get their 172s. The few remaining Crosscountry two-car 170/5s could then go to Scotrail, and release a few 158s to EMT to operate the Leicester-Stansted service.
I'm not sure that I'm completely happy with the idea of splitting the service, as I know a few people who travel from Birmingham-Stansted by train to catch the larger flights from Stansted that Birmingham doesn't cater for, and I know others who live in East Anglia and find the Cambridge-Birmingham service extermely beneficial for avoiding the dreaded A14. But it does seem like quite a convenient break-down of the route.