A mutt may be healthier than a pedigree. But a cross breed may inherit the worst health problems from both sides.
That depends what the 'worst health problems' are. If they are recessives carried by only one of the two breeds used in the cross - something like Collie Eye Anomaly, say, which is carried by a handful of breeds - the puppies might be carriers but they won't be affected. And a mutt will be much less likely to be affected by health problems resulting from extreme anatomy: a pug x JRT will almost certainly have better breathing than your average 100% pug.
One reason that health tests are such a thing with pedigree dogs is precisely because they are pedigree dogs: they are highly inter-related and so genetic nasties that would have only rarely popped up have become endemic in some breeds due to, for example, the overuse of popular sires.
I'm struggling to see that if you have a pet dog, do all the health checks on both sire and dam, want a puppy out of your bitch and breed from it, doing everything right from feeding them correctly, to using the vet if necessary, to having a number of people who want a pup out of your bitch, you are doing anything particularly terrible.
Well, precisely. And I know pet owners who have done all of this with the exception of health tests on the dam, because the health problems they were concerned about were primarily recessives and the sires were tested clear.
People are going to get puppies from somewhere. They can get them from a puppy farm, or they can get them from show or working breeders, or they can get them from people who breed their pets. If you're buying a puppy, you need to do due diligence on the breeder, whoever that breeder is or claims to be. I wouldn't buy from anyone without first checking the COI of the litter: show litters can have terrifyingly high COIs. You need to be alert to the signs of a puppy farm (including renting a house as a front). If you're satisfied on that score, ask more questions. If it's first generation cross of two breeds with dodgy hips, ask to see test results. And so on.
Nobody is saying that it's okay to sling two dogs together and see the ££££ signs. But it's not a sin to breed your pet if you do it sensibly.