I feel like you've misunderstood the papers, and what the motivation behind them is. The argument isn't that Intel chips are more efficient, it is that these schemes simply don't require continuous computations.
As for the two schemes, you are correct about the first paper that it depends on the timeouts enforced by SGX. However, the second paper makes no such assumptions and doesn't even assume that the enclave is secure; all it relies on is the attestation service (which is remote).
As for the two schemes, you are correct about the first paper that it depends on the timeouts enforced by SGX. However, the second paper makes no such assumptions and doesn't even assume that the enclave is secure; all it relies on is the attestation service (which is remote).