The venue you refer to is not suitable for kids, has very strong currents, is incredibly tide dependant and mobbed by the locals when remotely good.
to the OP, I think you’ll struggle to beat Gower for the balance of surf and MTB. Cornwall is more likely to have waves, and the water is prettier (if colder), but the mtb isn’t as blatantly easy.
Isle of Harris. Probably the best beaches in the UK, West coast surf and some fantastic trails. Wish I had my surf board when we were up there this summer.
Difficult to call. Woolacombe is probably nearest for getting to exmoor, easily 40 min drive once you have wiggled to somewhere. I live in Okehampton, it’s 40mins to westward ho/bude , nearly an hour and a half round to woolacombe to be honest. These are not fast roads! Good Dartmoor is easy another 30mins from here.!okehampton is definateoy no good for riding!! Polzeath to cardinham is probably 40mins but it’s your only riding option.
I would just have a beach holiday and settle for a bridle path explore at max. Some nice bridleways around llangennith but afan is probably 45 mins away easy! The 2 activities don’t go together well!!
The ride to sandwood bay in scotland is lovely and it’s never flat!!! You can do both in the same day, everyday then!
UK in summer is always going to be hit / miss with surf. SW Kernow, W coast Ireland, Hebrides or North coast Scotland will be more likely to have swell but you can be lucky or not. That said we had cranking surf on the BH weekend in August, so you can be lucky. If it happens to be flat and theres no decent riding might be frustrating?