I have seen this question twice on MSE (here and here), but both questions were considered unanswered by the authors, and the hints there didn't really help me. So I decided to write a new inquiry regarding it.
My idea is to use the following fact to prove that $M$ is totally umbilical, from which the result would follow:
"If $\alpha$ is a planar geodesic which is not a line, then $\alpha$ is a line of curvature"
If $p \in M$ is an arbitrary point, then, given any direction $v \in T_pM$, there exists a unique geodesic going through $p$ with tangent vector $v$ at $p$.
The problem is the part in bold in the above result: if this geodesic is not a line, then the direction is principal.
But what if, for some direction, the geodesic IS a line? How do I deal with that case?
If at least $3$ geodesics are not lines, there are $3$ principal directions, which already means the point is umbilical, I believe. But what about the remaining cases?
Thanks in advance