The original paper makes no such claim that most species arose recently. Here is what the authors say in a summary of their work.
“A straightforward hypothesis is that the extant populations of almost all animal species have arrived at a similar result consequent to a similar process of expansion from mitochondrial uniformity within the last one to several hundred thousand years.”
In other words, they did not find evidence of the first appearance or arising of these species in the last one to several hundred thousand years. Those species were already present for various periods of time longer ago than that.
What they actually found was this. They traced back from the genetic diversity found in the mitochondrial DNA of today’s species, using estimates of the mutation rate of this DNA, to a time when this DNA was uniform. From that method, they predicted the mitochondrial DNA of all of these species was uniform within each species one to several hundred thousand years ago. That does not mean the species arose at that time. It means the mitochondrial DNA of these species was uniform at that time. That uniformity could have been caused, for example, by reduction in population size of these species at the same time, such that members within each population had the same or a very similar mitochondrial DNA composition.
To say their findings mean most species arose at that time is a serious misreporting of the study results.