Is there an explanation for the difficulty of proving quadratic lower bounds for interesting NP problems? This is a follow up to my previous question: 

Best known deterministic time complexity lower bound for a natural problem in NP

I find it bewildering that we haven't been able to prove any quadratic deterministic time lower bound for any interesting NP problem that people care about and try to design better algorithms for. Our Exponential Time Hypothesis conjecture states that SAT cannot be solved in subexponential deterministic time, yet we cannot even prove SAT (or any other interesting NP problem) requires quadratic time! 

I know interesting is somewhat subjective and vague. I don't have a definition. But let me try to describe what I consider to be an interesting problem: 
I am talking about problems which more than a few people find interesting. I am not talking about isolated problems mainly designed to answer some theoretical question. If people are not trying to find faster algorithms for a problem then it is an indication that the problem is not that interesting.
If you want concrete examples of interesting problems consider the problems in Karp's 1972 paper or in Garey and Johnson 1979 (most of them). 


  Is there any explanation for why we haven't been able to prove any quadratic deterministic time lower bound for any interesting NP problem?
Here is an explanation from Lipton's blog: Bait and Switch: Why Are Lower Bounds so Hard? 

As Grochow observed, Rudich's insight applies equally well to $n^2$ lower bounds as to super-polynomial lower bounds. 

Rudich's insight explains why any lower bound proof that it is based on the following method cannot work. 


  "any computation that computes $f$ must make slow progress toward $f$. Each computational step can only get you a little bit closer to the final goal, thus the computation will take many steps."


Basically, there is no measure of progress that can survive the Bait and Switch trick of Rudich and successfully leads to a lower bound.