What evidence is there that plant evolutionary history "dances to a different beat" than that of animals?
I'm not going to give an extensive account on "mass extinction" epistomology here but I think the first thing one has to consider is the difficulty of studying numerically paleobiodiversity (mostly because of a gappy fossil record but also an historically unbalanced sampling effort and the rock availability for some of the time periods); although, along the years, we designed better tools to compensate for those biases (see the whole bibliography of John Alroy for instance), this is still a major issue.

Furthermore, the recognition of the "Big Five" (i. e. the five bigger mass extinctions of the Phanerozoic: End Ordovician, Late Devonian, Permian/Triassic boundary, Triassic/Jurassic boundary and Cretaceous/Paleogene boundary) basically derive from the work of Sepkoski (1978, 1979; Raup &amp; Sepkoski 1981, etc...) who studied the diversity of marine biota and more specifically (judging by Sepkoski 1981) marine metazoans. As a consequence everything terrestrial and every non-metazoan (plants, protists, bacteria) are not included in the classic recognition of those events. It so happens that at least the P-T and the K-Pg events have been also recognized in some marine protists (namely foraminifera) and in terrestrial vertebrates (dinosaurs for the K-PG and many groups of 'pelycosaurs' for the P-T to name the more famous); but conceptually this is a significant bias as it may happen that the geological event driving one of those mass extinctions can only affect marine organisms (oceanic anoxia being the most obvious example).

That's for the epistemology part. Now the plants themselves: as you noticed with your quote of Traverse (1988) it is thought that the Embryophyta (i. e. land plants) have not been affected by the Big Five. This idea have since then been somewhat challenged (e. g. McElwain &amp; Punyasena 2007) with evidences that plants indeed were affected by some of those extinctions but maybe not as drastically as what we think happened for marine faunas or terrestrial vertebrates.
It needs to be considered however that the earliest evidences for the existence of land plants date from the Silurian (see Kenrick &amp; Crane 1997 for a short review). Their fossil record remains extremely gappy until the Carboniferous, basically, meaning that the first two of the "Big Five", if they happened to have affected land plants, can not be observed.
Additionally Angiosperms appeared in the early Cretaceous and diversified significantly toward the end of the Cretaceous, meaning that if the K-Pg event had an effect on the other land plants, the number of extinctions in those groups would be drowned by the number of originations in Angiosperms (I don't have any reference for that, it is just food for thought).

Finally, land plants are not the only group that doesn't dance to the beat imposed by the Big Five. For instance there is evidence that radiolarians (a group of marine heterotrophic planktonic protists with an extensive, continuous fossil record) were not affected by the K-Pg crisis (Hollis et al. 1995) nor by the T-J event (Kießling &amp; Danelian 2011). There are still thought to have been, to some extent, affected by the P-T but there is no clear evidence (to my knowledge) for them being affected by the two previous crisis either.
And of course there is the multitude of groups with no (or minimal) fossil record (because they have no fossilizable parts for instance and thus are only recorded in extremely rare occasions) for which we have literally no idea of what happened to them.

References:
Hollis C., Rodgers K.A. &amp; Parker R.J. 1995. Siliceous plankton bloom in the earliest Tertiary of Marlborough, New Zealand. Geology, 23: 835-838.
Kenrick P. &amp; Crane P.R., 1997. The origin and early evolution of plants on land. Nature, 389: 33-39.
Kießling W. &amp; Danelian T., 2011. Trajectories of Late Permian – Jurassic radiolarian extinction rates: no evidence for an end-Triassic mass extinction. Fossil Record, 14: 95-101.
McElwain J. C. &amp; Punyasena S. W., 2007. Mass extinction events and the plant fossil record. Trends in Ecology &amp; Evolution, 22: 548-557.
Raup D. M. &amp; Sepkoski J. J. Jr, 1982. Mass Extinctions in the Marine Fossil Record. Science, 215: 1501-1503.
Sepkoski J. J. Jr, 1978. A kinetic model of Phanerozoic taxonomic diversity I. Analysis of marine orders. Paleobiology, 4: 223-251.
Sepkoski J. J. Jr, 1979. A kinetic model of Phanerozoic taxonomic diversity II. Early Phanerozoic families and multiple equilibria. Paleobiology, 5: 223-251.
Sepkoski J. J. Jr, 1981. A factor analytic description of the Phanerozoic marine fossil record. Paleobiology, 7: 36-53.
Traverse A., 1988. Plant evolution dances to a different beat. Historical Biology, 1: 277-301.