i definitely think she could contain a tiger's eye
as for why she's called agate, one of her components may just be an agate, i don't buy into the whole "name themselves" thing (not everything rebecca says is finalized, the podcast is just fun banter and isn't meant to be taken as fully canonical information)
and also, while yes agates are commanders, they are not bigger than other quartzes, we've seen two singular agates and they are not big