Are there published mathematics research papers that contribute a new approach for understanding a known result, rather than new theorems?
There are many papers out there that give a new proof of a previously-existing theorem. You can search for "a new proof of" in mathscinet (or even google) and find lots of good papers in respectable journals.

However, the new proof must provide new insight to be worth publishing. Two proofs can look different but actually be very similar, in the sense that they are relying on the same key ideas. This can be a bit subtle (and even subjective in edge cases), so it's a good idea if possible to consult with an expert in your field before making your findings public.

But if you manage to prove the result in a more elementary way than the original proof, that's a big point in your favor. It's always good to know that a result can be proven without appealing to a powerful theorem. If you go in the opposite direction (e.g. showing that a theorem from the 1950s also follows from Fermat's Last Theorem), you're less likely to impress people.