Is the prayer considered valid if the worshipper begins the Eid al-Adha prayer after six additional Takbeers, besides the opening Takbeer, due to severe crowding?

Your prayer is valid, and your takbir was correct. If a latecomer joins the Imam after he has pronounced the takbir, the latecomer should also pronounce the takbir, especially if he hears the Imam reciting, in which case he should listen attentively. If he is far away, he should pronounce the takbir. And whoever catches the recitation with the Imam should pronounce the takbir.