The only real benefits I can see via (and calling at) joined up Bradford are direct trains from Ilkley to Manchester / leeds, S&C / Airedale to manchester / leeds. (and vice versa). All would involve using diesel trains or bimodes.
All above nice to have, but cost makes above irrational in my view.
I was going on the OP's suggestion that seemed to be a chord from the Pudsey direction to the Shipley direction (but not any connection form the Halifax direction?)
Send the Manchester trains via Shipley triangle
That's going to be a couple of extra trains per hour through Shipley and down the Aire Valley (i.e. the half hourly Forster Square stoppers replaced by the two Manchester trains, one Blackpool and one Halifax service) - not going to be easy!
There's much criticism of the Crossrail idea, basically that Bradford and its satellites aren't important (like Leeds is) and can't generate traffic - not the view of the LYR or the MR or of most other ambitious towns and cities of its size
If you were building from scratch then, sure, you'd probably have a through station
But, given that we have the infrastructure that we do, it'd cost many millions of pounds to create a through station, so what benefits would it bring?
Not a huge amount to Bradford, since it already has links to all of the places - there's the benefit of being able to head to one central station and know that all Leeds services will depart from it, but it won't give Bradford any additional services to anywhere that it couldn't already have
Not to Halifax and the wider Calder Valley, since their services to Leeds would now take the slower route via Shipley
Ilkley and Skipon would have some new links in theory, but any service from Ilkley or Skipton to Manchester would presumably have to replace some of the current Leeds services towards Halifax/ Hebden Bridge etc
I suppose you could run the Lancaster/ Carlisle services via Bradford to improve it's connections, but by significantly slowing down the services from Leeds to those places, so what'd be the point?
If you can build it for a fiver then, great, I'd back it. But if it's going to be hundreds of millions of pounds then I'm struggling to see what actual benefits it would bring (compared to, say, wiring up the current line through Bradford Interchange)