What's the alternative should PKI collapse? Premises - please check it and critique if necessary:

If factoring the products of large numbers suddenly becomes simple, PKI goes up in smoke.

Is this correct? 

If it is correct, it seems to me that the only alternative is a return to shared secret schemes… is there anything serious out there that still relies on shared secret (and not in a hybrid fashion like SSL)?
No, this is not correct.

Your fallacy here is to assume that a public-key infrastructure implies RSA. RSA is just a way of implementing an PKI.

There are other cryptographic schemes out there that do not reduce to the problem of factoring large numbers.