Chickens are an old-world animal, domesticated from the red junglefowl of Southeast Asia. IIRC there’s some possible evidence for chickens being taken to South America by the Polynesians, but they certainly didn’t become widespread in the Americas until the European colonizers showed up.
Maybe Native Americans ate the eggs of other birds that they did have access to, such as turkeys? But even if they did, it’s chicken eggs that are the ones easily commercially available in the quantities a restaurant would need, so…
You’re not wrong.
However, it is worth pointing out that the documented “smallpox blankets” stuff happened in the 1700s and 1800s, which was already a century or two after the continent had been greatly depopulated by diseases spread unintentionally.