Final answer:
The Omomyids were the primates that most closely exploited a tarsier-like niche, being small-bodied, nocturnal, and insectivorous, and adapted to a life of vertical clinging and leaping.
Step-by-step explanation:
The primates that exploited a tarsier-like niche, which includes being small-bodied, nocturnal, insectivorous, vertical clingers and leapers, were the Omomyids. The Omomyids were part of the larger group of primates known as the Eocene primates, and they shared many characteristics with modern tarsiers, suggesting a similar ecological niche. They lived during the Eocene epoch and were believed to be small, arboreal, and nocturnal, similar to the tarsiers of today.
Unlike the other primates mentioned such as Adapids (which were more lemur-like), Hominoids (which include great apes and humans), and Aegyptopithecus (which resembled Old World monkeys), Omomyids best fit the description of being similar to the modern tarsier with respect to their ecological role. This is supported by the fossil record that suggests they were part of the early adaptive radiation of primates that had evolved to exploit various ecological niches including the particular one in question.