In my experience it goes on all the time. Starting with certain hypotheses can make it very difficult to prove a theorem. Starting with a different definition can often yield an easy solution.
An immediate example is to prove that $e^x$ is differentiable. You can start by defining all exponential functions for the integers, then for the rationals. Then you can bootstrap your way to the reals, if you are a little careful, and you will get continuity. Now try to prove they are differentiable.
If you start with the definition of a derivative, you will quickly run into problems. I have found/developed 3 different direct proofs of this differentiability, none of which is really easy.
However, by turning things around, we can have a very easy proof indeed, and avoid all the detailed fiddling that is needed if we define exponential functions directly. To do this define
log x = $\int_1^x (1/t)dt$ for $ x \in (0, \infty)$
From this definition you can immediately and simply prove the usual properties of the logarithm. By this definition log x is monotonic and differentiable.
Then define $e^x$ to be the inverse of log x as defined above. It is immediately monotonic and differentiable because log x is; and will inherit from log x the usual properties of exponentials. You have also defined e as being the base of this exponential function, without an elaborate derivation of the number or what it may mean. Showing this e matches up with our usual idea of e is pretty trivial.
Thus a complex proof with the direct approach is reduced to a few simple lines with an indirect approach.
This is a simple example, but some of the best proofs I have seen began by turning the problem on its head.