The “fits and starts” model was first postulated to counter-act the lack of smooth progression of organisms that Darwin claimed should be in the fossil record. It is a model that requires the absence of data to be valid. Unless, of course, you can show “fits and starts” happening repeatably in the present.
There was some moth in England that was mostly white, but which then became mostly black, when the industrial age coal dust pollution made "mostly black" better camo. Then when that was cleaned up, it went back to "mostly white/gray". This was not adaptation by individual moths, but multigenerational changes to the species. The same effect is seen when bacterial species develop immunity to antibiotics. Those that happen to have better resistance, or better camo for the moths, live and reproduce, while the others die off and/or are eaten.
Sometimes the species doesn't survive the transition. Dinosaurs for example. The problem is that the process generally takes several generations, sometimes more than merely several, and in those animals that have lifespans meansured in years, that can take a long time, too long to be considered "in the present".
The whole process is no mystery to animal breeders, who provide both the culling and the increased reproductive success for the "desired" variations. But those are generally relatively long lived species, and the best that people can do is change traits. Creating a whole new species would take much longer, and would require a large but isolated population.
Dad used to take our culls to "underprivileged" senior citizens, who ate them. Mom refused to cook pigeons. :) (When he had chickens, the culls did get eaten.)