I'll try a partial answer to your question, but I think it should be enough for your quest.
Being NP-complete resorts to being NP AND NP-HARD.
Being NP-HARD means every problem in NP could be translated, via a polynomial time transformation, into this problem.
SSP is know to be NP-HARD (and NP-COMPLETE).
This means, for instance, that you could look-up in the internet, for the specific Polynomial-Time transformation that SHOWS that SSP is NP-HARD (on any paper that proves it).
That transformation is an "honest" algorithm, is guaranteed to run in Polynomial-time, and is written in some paper, somewhere. Let's call it 'SSP-T' (for SSP transformation).
The only way to prove that P=NP is to EXHIBIT an polynomial-time algorithm for a problem in NP.
So if you assume that P=NP, you are assuming YOU HAVE IN YOUR VERY OWN HANDS an "honest" polynomial-time algorithm for solving one (any-one) NP problem.
Let's call that algorithm: 'H'
Now... given any problem in NP, let's call it 'My-NP-problem'...
The Solution you are looking for is:
Apply SSP-T to transform it into an instance of SSP,
Now use SSP-T again to transform this SSP instance into an instance of 'H' (the ONE --any-one-- problem NP that YOU KNOW how to solve in P --based on the assumption that P=NP--),
Run H to find a solution.
Use SSP-T to interpret the solution under SSP
Use SSP-T once more to interpret the solution under 'My-NP-problem' (the arbitrary problem you wanted to solve in the beginning).
And there you go!
These 5 sequential steps are the "honest" algorithm you were looking for.
Each step runs in polynomial time and is guaranteed to exist by the very definition of each concept.
You could have chosen any other NP-HARD problem, instead of SSP,
because the definition of NP-HARD is the one that guarantees (by definition!) that:
In fact if you look the most usual way to show that any problem is NP-HARD is VIA a transformation into SAT-3, which was one of the first problems to be shown NP-HARD (sometime in the 60s/70s).
Let me know of any weakness you might find in the reasoning/explanation I gave.