I think that testing *is* part of the answer. More information is almost always good. Testing is not the problem. It is what we do with the test results that can be the problem.
As you point out, eliminating dogs from breeding programs only makes a small gene pool even smaller, which leads to new problems down the line. Sadly with regards to health testing, I see breeders "fault judging" - eliminating dogs because of one poor result. No dog is perfect! Many breeders only want to breed to the "best dog" (so we get the popular sire syndrome) or follow a philosophy of "only breed the best to the best". This is not how nature works! (And if we humans followed such a breeding philosophy, many of us would not be around!) Rather nature does lots of breedings ... but only removes the very worst. Over time, there is slow improvement *and* genetic diversity is maintained.
Health testing allows us to avoid matings that have a high risk of producing a dog with a serious defect - for example primary lens luxation (PLL) or degenerative myelopathy (DM). Health testing and record keeping allows us to reduce the number of dogs with a polygenetic problem in a pedigree - which is the best that we can do right now about polygenetic problems - for example hip dysplasia, upper airway syndrome, epilepsy, liver shunts, etc. Health testing allows us to be warned about things that might affect a dog's life ... such as macrothrombocytopenia in Norfolks terriers or portal vein hypoplasia in Norwich terriers.
It we do not test, then we do not know. If we do not know, then we are playing Russian roulette with regards to the health of our dogs and puppies bred.
I think health testing is a good thing ...