Simple answer to your question is 'yes', the C2C route from Rowley Bank to Sunderland is all either cinder track, hard-pack, or asphalt, so would be fine on 23mm tyres, if a bit uncomfortable for some stretches.
However, Rowley Bank is in Castleside, almost Consett, which is about 14 miles from Stanhope.
It's the bit from Stanhope up to Parkhead Station that you need to be careful on.
Simplest way is use the B6278 to take on Crawleyside Bank to Parkhead - it's all road, so no worries apart from the gradient.
There's a track of sorts beside the road to the east, but it's not suitable for 23mm tyres.
From Parkhead, it's 10 miles of plain sailing to Consett in all senses, all down hill and all on well-made or asphalt track.