It turns out that's correct, but there are plenty of other cycling bans without being designated a "special road" AFAICT. I looked for special road regulations for the A1203 (Limehouse Link tunnels - CS3 is the cycling alternative), the A45 (Nene Valley Way - NCR539 alternative) and A14 (west of Welford Road - no signed alternative) and didn't find them, for three other examples. I realise all of those have arguable reasons for banning cycling which don't apply to the A24, but I still think it is legally possible without special road status.
Another Scottish example
here which was rather controversial when introduced, mainly due to the lack of a suitable alternative, though I wouldn't want to cycle on the A90 anyway. This part of the A90 is not a special road.
All classes of traffic are permitted to use all-purpose roads unless specifically banned by a traffic regulation order, which is what happened on the A90.
No classes of traffic are permitted to use a special road unless specifically authorised, in the case of the A720 by the statutory instrument used to allow its construction. So technically it's not a case of cyclists being banned from the A720 but one of motor vehicles being allowed.
Non-motorway special roads are quite rare. Other examples are the A55 in north Wales and the cycle tracks across the original Severn Bridge.