The value of a proof is not only its conclusion but also the insight that it provides through its method.
The goal of mathematics is not to prove as many theorems as possible but rather to gain an ever deeper understanding of why certain statements are true. The way that something is proved can be more or less useful to advancing that goal.
As an example the elementary proof(s) of the prime number theorem are just about as famous as the original proof. Sometimes the second bite of the cherry is even juicier than the first.
Tbh, elegance is something programmers should strive for too. Elegant code is easier to build upon, easier to read/understand, easier to modify, easier to adapt. For all the same reasons mathematicians want elegance. Though it's true for many domains. People love to throw around the term "first principles" but that's not something you (usually) start at, that's something you derive. And it's usually not very easy to figure out
Ultimately, a proof is an argument that something is true. The simpler "more elegant" proof is generally going to be more convincing.