A recent question made me think about an obvious approach for circumventing the "algorithm is allowed to do anything" problem, when proving lower bounds. Instead of starting with a simple looking NP-complete problem, start with a powerful looking logical deduction system.
Then try to find a reasonable general term expression problem, such that equality of the terms can be decided by the logical deduction system, and inequality of the term expressions is NP-complete. (For example, Boolean algebra term expressions would work.) Then start to prove that the logical deduction system cannot efficiently prove equality of specific term instances, unless it has access to specific higher order features, like the cut-rule.
And then the goal would be to push up the required higher orders features as far as possible, i.e. beyond the cut-rule towards fully impredicative higher order logic. But already proving that you need the cut-rule would seem like a nice achievement. How far has such an approach to explaining NP!=coNP actually been pushed in the past?