Birmingham airport is better as an airport and Crewe is better as a rail interchange for most people coming from the West Midlands. I can't see there being any great demand for trains to Manchester airport from Birmingham.
Whilst I'd agree, there are a sizeable number on here who are convinced that Newcastle/ Edinburgh/ Glasgow/ Leeds etc all need regular services to Manchester Airport (despite them all having their own airports too!).
Whilst a significant number of XC passengers change at Birmingham, there are also a significant number who stay on, and I'd be concerned if all the Manchester to Bournemouth through trains were stopped
Part of the problem here is that we are all arguing "blind" - nobody knows the percentage of passengers who change at New Street versus those who remain on the train (and the percentage of those remaining on the train who would be put off by a change).
Liverpool lost its services through to Bristol/ Bournemouth (etc) some time ago, as did Carlisle/ Preston - but there are always going to be more places north of Birmingham than south of Birmingham (on the XC map) and there's no way that you can squeeze all of Liverpool/ Carlisle/ Newcastle/ Nottingham (etc) onto the few paths south of New Street.
Even if there are a "significant" number of people who remain on the train, there's no need for every service to run through.
If we accept that a Birmingham - Manchester - Birmingham diagram is four hours long (i.e. with layovers) then that's eight diagrams. We could run six of them with long EMUs (to provide a high capacity service on most journeys) plus one diagram to Exeter and one to Bournemouth - thus retaining a four-hourly through service for the passengers who would want a through service. That's also six Voyagers freed up to provide additional capacity onto other XC services.
XC used to have bi-hourly "through" services in the Virgin days (e.g. of the two Newcastle services, it was 3tph to Bristol and 1tp2h to Reading), so it's not beyond the realms of possibility to rejig the services a little to accommodate two four-hourly through services.
XC already have their "hot spare" at New Street, so you could deal with the additional platform space taken by interworking the "hot spare" Voyager with the other services (so that there was always one Voyager at New Street, but it doesn't have to be the same one all of the time).
Even if you had to diagram the remaining services to run Bournemouth - Birmingham - Exeter (diagram, not advertise!) three times every four hours then you'd still be freeing up half a dozen Voyagers for the busiest XC services and providing a lot more seats on the journey between two of the three biggest urban areas in the country.