We've got scores of Cockers at Hearing Dogs amongst other breeds and we don't find them to be intrinsically any unhealthier than our Labs, Miniature Poodles, Golden Retrievers or the Cockerpoos/other crossbreeds. Each breed/type has things that they tend to be more prone to but none of them are what we would deem unhealthy breeds or we wouldn't use them - we stopped sourcing Flatcoats and Cavaliers for this reason. If anything thinking of recent experience we've had more issues crop up with our Labradors and Goldies. The things we tend to see in our Cockers here are the usual - ear problems, the occasional eye problem (ectropion etc.), skin allergy or sensitive stomach. Our working Cockers are no healthier or unhealthier in general than our show Cockers, and we have more or less even numbers of each.
I've got a working Cocker who is an absolute crock health wise - Tilly is 8 and she has IBD, extremely severe osteoarthritis, IOHC, spondylosis and chronic ear infections, all of which is almost certainly immune mediated - but although I don't think that the working strain is immune to health problems nor do I think that the breed as a whole is riddled with health problems. As a point of interest although she's a rescue I've got her pedigree and I put her parents into Mate Select when it first launched; her COI is 5% against the breed average of 9% so although it is a useful tool it's not a guarantee that you're going to get a healthy dog, sometimes you're just unlucky which I personally think is the case with Tills.