If you could see the hassle involved in getting a frequency then, yes, we do appear to be short of spectrum. I can't see any reason why the CAA would invent these difficulties if they do not really exist.
You may have provided a bit of an answer - or at least a way to focus on flight safety issues - in your post. If you are unable to obtain the info you need from ATIS, then ask whichever unit you are speaking to to get it for you and explain why. A few reports going in to CAA from ATC will soon focus minds - and hopefully there will be a solution in there somewhere.
I wouldn't normally advocate increasing ATC workload - I've been that one-armed paperhanger too often myself - but if you need the info for reasons of safety, that's one of the things ATC is there for (although I admit that some of my younger colleagues do seem to hold a slightly different view!).