Of course I googled the problem. Most of the posts on sites like Backyard Chickens suggested heart problems or circulation problems. I worried that he might die of natural causes (which would be fine) but that the meat would then be wasted, especially if he passed away in the night or midday when noone would find him for a while. So we called Michele's friend Marvin who is a sustenance hunter. He agreed to put the poor guy out of his misery and then make use of the meat in exchange.
I feel badly that I don't have the strength to cull the birds myself, but I hope that whatever made the rooster suddenly fail won't affect the meat at all. Marvin will let me know, so I'll post the answer then.