Will mathematics journals accept a non-novel theorem, but with different approach

There is more to mathematics than theorems. Sometimes the best part is the methodology of the proof, especially if it can be used to also solve important problems in the field. If everybody has been thinking in a certain way about a class of problems for a hundred years and you give them a new way to think about it, you have made an important contribution.

So yes, a journal would publish that. But your paper needs to be clear about the novelty and importance of your approach.

My own dissertation had interesting theorems, but was noted for the proof of one of them that was something entirely new and unexpected.

Edward Witten not only published a paper of this kind (entitled 'A New Proof of the Positive Energy Theorem'), but the paper strongly contributed to getting him a Fields Medal, the highest award in mathematics. Both proofs in the paper had already been proved by Schoen and Yau using different methods: the key is that he was using new methods to carry out the proof.

Edit: I should also add that general mathematics journals often like to take interesting or easier new proofs of old theorems.

Later: I have thought about this a bit more and the truth is not as quite as glib as I suggested. Unfortunately these things are never black and white. The proof by Witten of that theorem was new but also interesting and surprising. Producing yet another proof of some workhorse theorem which people use on a daily basis might be a bit hard to publish (and I'm speaking from experience as well).