How to contribute to friend's mathematics paper without overshadowing? My friend has an early draft of a math result he would like to publish.

In telling me about his results, I made an observation that significantly simplified a portion of the proof. He wanted to list me on the final publication (not as a co-author, but more of a shout out for contributing), which I was happy with. I don't plan on going into academia so publications aren't essential to my career.

However, as I've thought more about the results, I've seen more and more places where the proof could be simplified using alternate methods, to the point where the proof can be reduced to a fraction of its original size and bears little resemblance to the original.

My friend doesn't know about the new results yet and I worry it would seem like all the work he had done before was unnecessary. But his work was certainly influential as it was easier to make the observations given that I knew what the result should be.

I certainly wouldn't want to withhold my findings, so I suppose my question boils down to how should I handle telling him?
Tell him what you have figured out.  Maybe he will want to modify his own proof and add you as an author.  Maybe he will just add a note that it has been pointed out to him that the result can also be proven using other specified methods.  Maybe he will look over your shorter proof with interest but not change his manuscript at all.

Understanding how a result can be proved is obviously of key importance in mathematics, but do not forget that result that he has proved ought to have some interest unto itself.  It can be more important to know that something is true than to have the most elegant proof of its truth, and realizing that a result is provable and interesting is a significant accomplishment on its own, even if a "better" proof can be found later.