Nine years is a long time, I can see why you would be reluctant to change anything on a tried and true system. Is this the first time you've had real trouble with cyano? (Other than just the occasional outbreak)
The old axiom "if it ain't broke, don't fix it" springs to mind but something's gone awry here.
I'll assume you checked all the obvious; all the ph's in the tank are working, you've got one directed to sweep across the bottom (higher flow makes it harder for they cyano to get a foothold), you're alk levels are good, the lights aren't old. You've already cut back on the feeding. Have you gotten the cyano to clear up at all in the six months and then it's come right back with a vengeance? If that's the case and all the above check out on the mark I think you may be right on the nutrient sink. However, if you've never gotten rid of it entirely for a time maybe you could try an all-out assault on the cyano and see what happens.
I'll also assume you did the blow it off the substrate/syphon it up/waterchange routine. Maybe you could give that one more shot, removing as much as possible and then leave your lights off for two-three days. Keep a good water flow over the substrate during this time and then turn your lights back on and see what happens. If you have a sump or if you don't have critters that would eat it up before it had a chance to do any good, you might want to try cultivating some macro and see if it can outcompete the cyano for some of the nutrients it's feeding on.
I know these are all pretty standard "cyano tactics" but sometimes a review of the obvious can make a connection with something else that might give you some insight. If your tank was a young tank I'd probably say go ahead right away with a substrate change-out but since it's so well established and I'll assume has mostly been trouble free up until now I hesitate to put your tank through the upheaval.
If you do decide to change-out the substrate you might want to "cure" some new sand by putting it in tubs covered with water from your tank's waterchanges. You could then seed it with some of the critters from you current sand bed and change the substrate out slowly. Or just get the new sand nice and populated before you change the whole thing over. Either way, it's going to be a pain but as CC is notorius for ending up as a nutrient sink it might be your only option.
Sorry for the long rambling post
~Alice