The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny factors of $J(C)$ are elliptic curves.

Now since over $\mathbb Q$ the isogeny factors of $J_0(p)$ correspond to galois orbits of newforms it seems to me that the question wether there is a largest prime $p$ such that $J_0(p)$ splits completely into elliptic curves over $\mathbb Q$ should be easier and I wonder wether the answer is already known.

I suspect that the answer is yes and that $p = 37$ is the largest prime such that $J_0(p)$ completely splits into elliptic curves. Indeed using Cremona's database of elliptic curves I verified that 37 is the biggest prime below 300000 such that $J_0(p)$ completely splits into elliptic curves.

The reason I only ask it for prime levels $p$ is that I already showed that if $p = 37$ is indeed the largest prime such that $J_0(p)$ completely splits into elliptic curves, then $N = 1200$ is the largest composite number such that $J_0(N)$ completely splits into elliptic curves.