I can see a case for a trained mathematician contributing a legitimately new perspective to the community by withdrawing and spending time looking at things in an unorthodox manner. That's closer to that Grothendieck was describing than what you're posing.

In the early to mid 20th century, it was feasible for someone to pick up a book on number theory and apply relative genius to an open problem to quickly solve it. The bottom line is that prerequisites for doing so were often just basic calculus and high school algebra. An understanding of sequences and series went a long way.

This isn't the case in the 21st century. We're firmly out of that territory. You can't offer a profound new perspective on a thing which you can't understand, and the barrier to understanding research mathematics continually rises. The forefront of modern mathematics is so far removed from even graduate level mathematics course material that it's not going to just be intuited through untrained brilliance. You have to actively learn it, which is (unfortunately) vanishingly unlikely outside of academia.

If the tools available to you are basic real analysis and linear algebra, P/NP is beyond your reach - full stop. At this point we actually have proofs that a valid proof of P/NP (and similar problems) cannot be achieved through large swathes of elementary techniques.

We don't live in a world like Good Will Hunting where an amateur can succeed by being a genius. That's useful in the long term but not enough on its own.