Can such results be published and how?
“Such results” certainly can be published, if by “such results” you mean genuinely new proofs of well-known theorems, especially if they contain a new and interesting idea (rather than being a trivial modification or variant of an existing proof), and especially if they are shorter than existing proofs, although that is not a necessary condition for a new proof to be publishable.
Such new proofs are published quite frequently. For example, when I was a graduate student I published a new proof of Stirling’s formula. It was published in the American Mathematical Monthly, a respectable journal that often publishes papers in this category.
Should I publish my result?
As others have said, you can at the very least write up your proof as a paper and submit it to arXiv to make it available to the community. And you can try to publish it in a journal - if it’s interesting, well-written and novel, I think you have a good chance of getting it published somewhere.
If so, as I'm not affiliated with any Research Institute how do I do that?
See this question for some suggestions.