Is there any evidence to back up this assertion? Seems to me that companies are voluntarily choosing to stay private, not that public equity markets can't handle the unicorns. (edit: Or is he saying that unicorns are overpriced because private investors value these companies more highly than public equity market participants would?)
All in all, business models that can't handle marginally higher financing costs probably weren't all that healthy to begin with. Rising rates should shake out the weaker hands, in a form of creative destruction.