"What if" history is always a difficult thing to write, but I would argue there are two key factors at work here.

First, Easter Island has no harbour and no significant natural resources. This made it both unattractive and unsuitable as a base for a western power which was interested in projecting sea power.

Second, the island went through a particularly unpleasant period in the mid nineteenth century when it was ruled by a French adventurer named Jean-Baptiste Dutrou-Bornier, known to Pacific islanders as "Captain Buttons" for the quasi-naval uniform he wore. Dutrou-Bornier was mainly interested in Easter Island's potential as a source of indentured labour (in practice, effectively slavery) for the guano mines then being exploited along the coast of Peru.

A number of Pacific islands were practically stripped of their populations in this period to provide indentured labour in Peru, and Easter Island was one of them. By the time that Dutrou-Bornier was murdered by the remaining Rapa Nui, the island's population had declined from around 10,000 at its peak to only 100.

Dutrou-Bornier did receive some support from the French in Tahiti during his seven-year "reign" on Easter Island, but – even by the remarkably low standards of this era in Pacific history - he was too unpleasant and too unprincipled a character to secure any sort of official backing for his rule over Easter Island; hence it was never added to the French Pacific empire, which might easily have happened otherwise. After his death, the practically deserted island was an even less attractive proposition for a colonial power.

Easter Island was instead "sold" to Chile in 1888. This last process set in motion by Dutrou-Bournier's successor, a British trader named Alexander Salmon. Salmon had married a Polynesian woman and was far more sympathetic to the Rapa Nui. Given the island's depopulated state, Salmon had little difficulty in acquiring practically of the land on the island and was - from his perspective, anyway - entitled to dispose of it as he saw fit. The purchaser that he found was the Chilean government, which formally annexed the island after purchasing Salmon's land.

Sources

Edward Dodd, The Rape of Tahiti: A Typical Nineteenth Century Colonial Venture Wherein Several European Powers with their Iron, Pox, Creed, Commerce, and Cannon Violate the Innocence of a Cluster of Lovely Polynesian Islands in the South Pacific Ocean (1983)

Steven R Fischer, The Island at the End of the World (2005)

David Hollett, More Precious Than Gold: the Story of the Peruvian Guano Trade (2008)

Jacques and Corinne Raybaud, Dutrou-Bornier: Le Capitaine Français de l'Ile de Pâques (2013)