Why Isaak and NOT Ishmael

I recently listened to a profound teaching about the subject of Isaac and Ishmael, and why Isaac received the birthright promise of Abraham, and not Ishmael.

Basically, the gist of the teaching was: Isaac was born after Abraham was circumcised, thus Isaac passed through the "cutting" (the circumcision of Abraham) when he was conceived, while Ishmael was already 13 when he was circumcised. There was some prep teaching about three different types of covenants in ancient times. Usually, blood was used in some fashion, not unlike modern tales where those entering a covenant will cut their finger tips, etc...

Personally, I'm not sure how I feel about the basis of this "proof" in favor of Isaac. I simply look at the promise made directly to Isaac's mother, Sarah, as proof enough:

Genesis 16:10 "And the angel of יהוה said unto her, I will multiply thy seed exceedingly, that it shall not be numbered for multitude."

The promise was not made to Hagar, Ishmael's mother, but to Isaac's. It can't be any more simple than that.

Shalom Aleikham.