That's a good point, but from the article that you linked they had been completely absent from the continent for up to 10,000 years, until they were reintroduced by the Spanish:

Equus flourished in its North American homeland throughout the Pleistocene but then, about 10,000 to 8,000 years ago, disappeared from North and South America. Scholars have offered various explanations for this disappearance, including the emergence of devastating diseases or the arrival of human populations (which presumably hunted the horse for food).

Despite these speculations, the reasons for the demise of Equus in the New World remain uncertain. The submergence of the Bering land bridge prevented any return migration of horses from Asia, and Equus was not reintroduced into its native continent until the Spanish explorers brought horses in the early 16th century.