I’m not particularly local to Clent, so haven’t ridden there much, but there is some decent riding there from what I remember. ISTR the riding at the Lickeys being nicer, but I’m sure somebody more local will tell me I’m wrong – it is certainly significantly closer to Birmingham City though.
As mentioned there is also a trail at Sandwell Valley, though that’s not in the direction requested by the OP (but possibly still closer to the south City centre than other options).
ISTR riding somewhere much closer in though it wasn’t thrilling, will see if it comes back to me…