Why are elliptic curves better than cyclic groups? The set of points of an elliptic curve over a finite field is isomorphic to the direct product of two cyclic groups (i.e. $E(F_{p^n}) \cong Z_{s} \times Z_{t})$.
What is the advantage of representing or conceiving the group as an Elliptic Curve instead of as a product of cyclic groups?
The first assumption: $E(F_q) \simeq Z_s \times Z_t$ is always true!! (see Handbook of Elliptic and HyperElliptic Curve Cryptography)
It is never said that coordinates of points lie on $Z_s \times Z_t$ which has no sense except if $s=t$.

But the isomorphism is not trivial to compute (it is in fact hard to compute, since it is related to the discrete logarithm for the left-to-right part).

And working only with the representation $Z_s \times Z_t$ will weaken the discrete logarithm so useful for cryptographic purposes.

In brief, if you can exhibit a curve (or demonstrate that there is a NIST curve) for which the computation of the left-to-right part of the isomorphism is easy, then this curve MUST be considered as weak and must never be used for any cryptographic purpose.

Concerning quantum computers, EC-DLOG will be solved as easily as for any cyclic group, since the Shor algorithm can be adapted to solve it, but don't ask me about details, I don't understand them...