No, for every number field $K$ and $n \in \mathbb{Z}^+$, there exists a genus one curve $C_{/K}$ such that the minimal degree $[L:K]$ of a finite field extension $L$ such that $C$ admits an $L$-rational point is $n$. This is a $2004$ theorem of mine.
Over function fields of characteristic $p > 0$, the same result holds, at least if you take $n$ to be prime to $p$. I think this result follows from the recent (solo) work of (my friend and collaborator) Shahed Sharif. (The result should hold in complete generality, and I believe Shahed and I already know how to prove it. But studying the period-index problem for $p$-torsion classes in characteristic $p$ in full generality gets rather technical, and we have put this aside for a couple of years now...) In any case it is much easier to see that the minimal degree can be arbitrarily large as one varies over all genus one curves $C_{/K}$ even with a fixed Jacobian elliptic curve $E$. For a proof of this -- and actually of a significantly more general result -- see Theorem 11 in these unpublished/unpublishable lecture notes.