Ugly master's thesis but one great proof
What would be a suitable way to get other mathematicians to know the proof? 


Publish it!  Write the proof by itself in a short, self-contained paper, and submit it to a journal.

There are lots of papers in the literature giving "A new proof of ...".  You may not be able to publish it in a very top journal, but if it's really new and significantly simpler than existing proofs, some reasonable journal should accept it.

Additionally, you can post it on https://arxiv.org, so that it's permanently available to the community, whether or not it eventually gets published.