Strongly NP-hard problems and Dynamic Programming
You're generalizing too much from a small sample size.  There is no guarantee that all weakly NP-complete problems can be solved using dynamic programming; I'd expect probably there are some that dynamic programming isn't much good for.

And, there is no guarantee that a strongly NP-hard problem will take exponential time to solve.  There are NP-complete problems that can be solved faster than exponential time.  (See also the exponential time hypothesis.)

Dynamic programming is just one algorithmic technique among many.  There's nothing magical about it, and it's not a be-all and end-all.  There are other algorithmic methods other than dynamic programming.