Darwin put forward the concept of evolution through slight modifications not because there was the overwhelming evidence to that but because he aimed his theory against the religious belief in creation of species at once in the final form. Slight modifications is the complete opposite of this concept. The rationale behind the idea of slight modifications was this: if big modifications are admitted then what is the limit to the size of such modifications and ultimately what would prevent species to be indeed created at once in the final form ?
Not that this is how Darwin explained why modifications have to be slight. (Actually he did not explain it at all.) But it is my hypothesis of what motivated him.
The evidence he actually had was weak (just slight modifications of some body parts of some species). Most other observable variations were just giant leaps. But they were supposed to be bridged by finding the fossils of the intermediary species that died out.
But there are other reasons of why modifications have to be slight, which Darwin definitely did not consider. This is the propensity of species to kill, expel, or abandon the offsprings that vary from them significantly. In either case the significantly deviating offsprings die. Still, it might not be applicable to fauna.
But if significantly deviating offsprings are born in significant numbers (i.e. in bursts) and this deviation transpires only after they reach the age when they can survive on their own even if expelled or abandoned by the parents, then such bursts of significantly deviating offsprings would not die. The article "New Species Evolve in Bursts" published in "Nature" in 2009 seems to support this hypothesis.