Where to publish a new proof of an old theorem?

This question, as stated cannot be answered. Everything depends on the theorem and on the proof, and this information you did not state.

For example, at least one Fields medal was awarded for a "new proof of an old theorem" (Selberg, 1950). A new proof can be published in principle in any mainstream journal, if the theorem is important and the proof gives an important new insight.

If the old theorem is something commonly seen in an undergraduate math class (with the old demonstration), then this might be appropriate as a "Note" in the American Mathematical Monthly.

What could happen if you submit it? They may publish it. The referee may give you a reference for it. They may respond in the same way as the Gazette.

What if the old theorem is not commonly seen in an undergraduate math course? When you write a textbook on that area of math, you can include your new proof. But if you think it unlikely you will write a textbook on this, then probably there is little prospect for publishing this. Maybe if you make it known to the experts* then some day one of them may include it in their new textbook.

*Perhaps by posting somewhere on-line...

Alternately, when you write a paper on a related topic (ie, which already develops the necessary machinery), you could perhaps include it somewhere in that paper? I've seen this done numerous times.