Could giving an alternative approach to proving an established result be worthy of publication?
Often enough the proof of a theorem is more important than the statement of it. This is especially true when a new proof gives some insight into a problem that the original proof did not.
I don't know whether your new proof is different enough or interesting enough or gives new insight, but if it is, then it would be an important thing to publish.
An example of such a problem is the Four Color Theorem. When I was in school it was the Four Color Hypothesis and nearly everyone in my age cohort thought about it for a while I suspect. It was originally proved using computers to examine a very large number of special cases. But to a mathematician, this is a very unsatisfactory proof. One would love a short and insightful proof not depending on computation as it would be very likely to give insights into the nature of planar maps.
The reason that such insight is important is that it can often lead to solving additional, unsolved, problems.
So, you can certainly submit such a paper for publication, but it is editors and reviewers that will decide whether it &quot;can&quot; be published. But the feedback you get will be valuable in any case.