Why can’t we know the specific outcome of an exchange economy with initial resource endowment?
The idea of the contract curve is just restricting the Pareto set so that no one player is worse off than the initial allocation. There's no concept of "price" involved here. When your endowment is $(x, y)$ and you're asked whether you'd like to get another bundle, $(x', y')$ instead, you only check if your utility is higher. If it is, then you'd take it.

The concept of price comes in with the Walrasian equilibrium (p. 654 of MWG) which is yet another level of refinement on the contract curve (i.e. the "core").