elliptic curves - Hecke operators acting as correspondences?

The answer to your last question is yes, modulo my own misunderstandings. In Scholl's "Motives for modular forms", §4 (DigiZeitschreiften), the Hecke operators are defined in this way, and I think the equivalence of these definitions is implied by the diagram (3.16) and proposition 3.18 of Deligne's "Formes modulaires et representations l-adiques" (NUMDAM).

Let p be a prime not dividing N, and let Y(N,p) be the moduli scheme parametrizing elliptic curves with full level N structure and a choice of a cyclic subgroup C of order p. There are two natural "forgetful" maps q₁ and q₂ from Y(N,p) to Y(N) -- the former forgets the cyclic subgroup, the latter takes the induced level N structure on the quotient by C.

From these two maps, one gets two different families of elliptic curves over Y(N,p). Explicitly, the pullback of Uⁿ(N) along q₁ is isomorphic to the n:th fibered power of the universal elliptic curve over Y(N,p); let us denote it Uⁿ(N,p). Let Q(N,p) be the quotient of U(N,p) by the cyclic subgroup C, and take its n:th fibered power as well. Then similarly Qⁿ(N,p) is the pullback of Uⁿ(N) along q₂. Finally, the quotient map gives us $phi : U^n(N,p) to Q^n(N,p)$.

But this data gives us the right correspondence on Uⁿ(N) over Y(N), namely, one takes the composite $q_{1ast} phi^ast q_2^ast$ (where I use q₁ and q₂ also for the induced maps on fibered powers of universal curves).