Could an NP-Hard problem be in P in after a basis transform?
The only way they could propose to solve the problem (that I can think of) ... would seem to be exponentially hard."


That's your core fallacy, and "that I can think of" is the answer. The complexity of the problem did not change, but the transformation made finding an efficient algorithm easier (as a tool). 

Yes, it's certainly possible that we have yet to find a similar tool that, once we have it, allows even our feeble minds to easily find efficient algorithms for NP-hard problems. That would be a constructive and generalisable proof for P=NP, that is something like a holy grail.