Crossbreeding only reduces genetic conditions when the dogs are genuine Heinz 57s, .
Actually, no.
Some breeds have a lot of dogs that carry one or more recessive conditions. A carrier (one copy) will show no symptoms. But if it is mated with another dog that carries the same recessive, on average a quarter of the puppies will get a double dose and develop the condition. This why responsible cocker and springer spaniel breeders will test for PRA and only mate carriers to non-carriers, and if all else is equal, elect to breed a dog who isn't carrier rather than one who is.
There are also a couple of conditions which are pretty much breed-specific, like wolfhound rhinitis.
You could breed a carrier wolfhound to just about any dog on the planet that wasn't a wolfhound, no health tests, and pretty much guarantee that no puppies would be affected.
Similarly with PRA: quite a lot of breeds don't carry PRA (or only vanishingly rarely), so you could mate the dogs without health tests and not worry about the puppies going blind in later life.
There will be illnesses that are genetically mediated that we haven't spotted yet or have spotted but can't test for. And then there is the whole issue with the immune system: the narrower a genetic pool an animal is drawn from, the more likely (likely, not guaranteed) it is to have a compromised immune system. It may be this that underlies the issue with cancer in flatties, or that maybe written into their genetics - there is research on this underway. In any case, a poor immune system will predispose to allergies.
There are no guarantees and even a fairly random-bred dog can have issues (terriers are known for itchiness, and our not-sure-what's-in-there-call-it-a-JRT was on the itchy side) but you up your odds.
But yeah, if you cross-breed dogs from breeds that are both known for PRA or bad hips without health tests, you are taking a big punt with the welfare of the puppies. But they'll probably at least benefit from decent immune systems.