No, a dog and a wolf are both equally viable = able to live.
Moreover a dog is better adapted through selection of genetic traits towards its specific function.
Run a wolf in a greyhound race. Who wins?
Run a wolf in a dogsled race against huskies. Who wins?
Try to get a wolf to herd sheep as well as a herder. Who wins?
Try to get your wolf to dig a varmit out of a tunnel as well as a dachshund. Who wins?
Yes, the vast majority of all species that ever existed are now extinct. You think that they went extinct through mutation, but we did not? What protected us and every other species extant upon this Earth from this same ‘great plague of mutation caused extinction’ you propose?
This should be amusing!
Sorry but the wolf has more genetic viability than any dog breed out there. Surely you know that their DNA deviates less than 1% yet there are many many attributes in the dog’s DNA that have been turned-off and can not be reversed w/o much inter-breeding w/ the wolf. How about you take just 2 dachschunds and breed them back into wolves.
We do have mutations, all living things do, but the survivors obviously have not accumulated enough genetic drift to be driven to extinction. I’ll grant you that a few species have gone extinct due to negative changes in their selected environments but certainly not all.