If all NP-complete problems can be converted to 3-SAT problems I had an idea that would not solve the NP problem but might be a practical solution.
What you could do is simply, make a huge table of 3-SAT problems and whether they are solvable or not. Store this on a giant server.
Then it takes less than polynomial time to look at a 3-SAT problem and look it up in a table.
Two questions, for practical purposes, how big a table would be useful? Does such a table already exist?
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You can't just brute force 3-sat. That would take too much computational resources. I am pretty sure all the computers on earth would not be enough. This is typical of all problems considered infeasable in computational complexity theory. The doubling of computational power that is happening due to Moores law still isn't good enough to solve many problems
I think your approach would be completely infeasible and I am sure that no such table exists. The good news is that SAT solvers using clever algorithms and heuristics are able to solve formulas involving thousands of variables with acceptable running times. If you try to estimate how many entries there would be in your table if it contained all formulas with up to 10,000 variables and 1,000,000 symbols, you will see why I say your approach is infeasible.
The moral is that just because a problem is NP-hard does not mean that algorithms to solve that problem are useless in practice.