> And that "we haven't built it yet", sans additional explanation, is pretty shoddy evidence for the non-existence of something.

It's not claimed sans explanation. Hypercomputation would break a lot of physics. Others have mentioned the Bekenstein Bound, but just consider just basic concepts like signal to noise ratios. You'd need infinitely precise detectors for hypercomputation, detectors capable of distinguishing literally any signal, not matter how small, among any amount of noise, no matter how overwhelming.

These properties simply stretch credulity, to say the least.