Why does the prover in an interactive proof run in exponential time with the space of the computation?
The whole point of verifiable computation is to allow the verifier to verify that the computation was performed correctly without performing the computation himself. Since as usual we associate "polynomial-time" with "efficiently computable", if the computation takes poly time there is no point in using verifiable computation, since the verifier can just as well perform the computation himself.

In proof systems in general, if the verifier can determine by himself the truth of the statement in question, there is no need for a prover at all.