The short answer is they can't, because gene variants we've selected for are lurking about in the genome.
All dogs, purebred to street cur, are distinct combinations of different versions of the same twenty-plus thousand genes. What makes one dog different from another, or almost the same as in the case of purebreds, is how the genes are expressed or restricted from being expressed.
In a wild world, breeds would disappear. There would still be different colors, sizes and even shapes, but extreme exaggerated traits wouldn't survive.
Would wild dogs be able to sustain a healthy population under intense pressures of natural selection? Some populations likely would. Half-feral developing world street dogs have done a pretty good job. But these dogs are not off-spring of modern companion dogs.
Geographically isolated for thousands of years, dog populations that have evolved by adapting to natural and cultural environments are likely more genetically diverse than modern breeds. That's why they can adapt relatively quickly in an ever changing world - by calling those variants into service.
But it's not clear if our modern westernized breeds still have the genetic wherewithal to adapt to natural conditions. We may have thrown all those genetic babies out with the bathwater when we selected for very specific and exaggerated mutations to make breeds.
So keep your babies inside when it's cold, and get the rain slicker out in a downpour.
Read more here about how wolves and dogs are different.