If the final product of an egg is a chicken, then why wouldn’t the egg itself be considered a meat product? Thank you for your time.

A complete egg, even it is found inside a slaughtered chicken, is not a meat product. This- even though regarding other laws, an egg still attached to the chicken is considered part of the chicken. If, for example, a person were to hit a live hen with a stick detaching an egg, it would be forbidden to eat the egg since the egg would be considered "ayver min hachai" [=a limb of a living creature]. The reason the egg is not defined as a meat product is because eating fowl and milk together is a rabbinical prohibition and the prohibition was not extended to eggs (see Shach Yoreh Deah 87 9).