> But if you tell them they can't do this profiling, then they'll presumably resort to some mix of (1) no discounts with slightly lower overall pricing or (2) some loss leaders but otherwise regular pricing.
That's not true. Competition with other market participants should in theory (assuming competition) be sufficient. The firms are in general already charging the profit-maximizing price. They could, however, increase profits with more information.
But there is the additional problem that they will also monetize this information by selling it into advertising markets. The whole point of this business model is to capture previously uncaptured value that would otherwise have been shared in the commons.