Is every point on an elliptic curve of a prime order group a generator?
This is true of any group of prime order, over elliptic curves or not. This is due to Lagrange's Theorem which states that the order of a subgroup $H$ of group $G$ divides the order of $G$.

Since orders are elements of the ring of integers and since this is a principal ideal domain, unique factorization exists and primes make sense. Or put another way, primes behave how you've been taught they do. Therefore the only possible subgroups of a group of order $p$ are the trivial group of order $1$ and the group of order $p$, i.e. the group itself. 

This has a consequence, usually presented as a lemma of Lagrange's theorem - no element can generate a subgroup, so it must generate the full group.

For elliptic curve groups, there is a valid proper subgroup, the trivial subgroup consisting only of the identity, in this case $\mathcal{O}$.