Does conservation of momentum really imply Newton's third law?

Right, you could satisfy the momentum conservation by forces that don't satisfy "action vs reaction" law $F_{ij}=-F_{ji}$ but the relevant formulae would have to depend on coordinates and momenta of all the particles. If you assume that the particles are controlled by two-body forces only, the momentum conservation does imply that $F_{ij}=-F_{ji}$.

It doesn't, and there are a list of examples in this nearly identical question: Deriving Newton's Third Law from homogeneity of Space

There are no examples of fundamental classical three body forces where the forces are contact forces and linear gravity/EM, because linear fields are two-body interactions. The most obvious non two-body force is in the strong nonlinear gravitational regime.

Other physical examples are things like nucleon-nucleon 3-body forces, which are unfortuately completely quantum.