Every once in a while some mathematicians publish (mostly on the American Mathematical Monthly) a new proof of an old (nowadays considered "basic") result in analysis (calculus).
This article is an example.
I would like to collect a "big list" of such novel proofs of old results. Note, however, that I am only looking for proofs that represent an improvement (in some sense) over standard alternatives which can be found on most textbooks.
Andrew Bruckner's survey paper Current trends in differentiation theory includes a lot of examples of new simple proofs of results previously having difficult proofs, but most of the examples are probably past the level you want. Probably more appropriate would be the use of full covers in real analysis.
(ADDED NEXT DAY) When I got home last night I realized that the Bruckner paper I was thinking about isn't the paper I cited above, but rather the paper below. I couldn't find a copy on the internet, but most university libraries (at least in the U.S.) should carry the journal. Nonetheless, the use of full covers in real analysis, which I've already mentioned, is about as close a fit to what you're looking for as I suspect you'll get.
Andrew M. Bruckner, Some new simple proofs of old difficult theorems, Real Analysis Exchange 9 #1 (1983-1984), 63-78. [Go here for the zbMATH review (Zbl 569.26007) of the paper.]