Cosford to Sutton Coldfield tickets are valid via Birmingham New Street, as you'd expect. However, in the direction towards Sutton Coldfield it is quicker to avoid New Street by changing at Tame Bridge Parkway and Aston, since Cosford to Birmingham stopping trains are now diverted to run via Tame Bridge Parkway. You can still change at New Street if you prefer, but it takes 10 minutes longer (at least, based on the official connections, which can often be beaten in practice).
In the direction towards Cosford, the connections work out a little bit differently and it is quickest to change at New Street if you depart Sutton Coldfield at xx49. You can get to Cosford at the same time (xx15 1.5 hours later) with an additional change at Wolverhampton but there's no real benefit to this. If you leave Sutton Coldfield at xx19 then you could also reach Cosford in around 1.5 hours by changing at New Street and Shifnal, but the double-back between Cosford and Shifnal isn't permitted on a Cosford-Sutton Coldfield ticket.
The site is correct it isn't valid via Birmingham New Street
Not sure where you are getting this from, but it absolutely is valid via New Street. Just not via Shifnal (whether you travel via Tame Bridge Parkway or New Street).