I’d say the artistic value is the main value in general —- if you know mathematicians they will often refer to a ‘beautiful’ proof, aesthetics is a critical part of math.
An early (easy to understand) proof that fits the bill for me is proof that the sqrt(2) is irrational: a very natural thing to think about —- for a unit square, what is the measure of the diagonal? And people were killed in ancient Greece over the proof, but it’s comprehensible with basic algebra.
IMHO it's groking that allows higher level metaphors and eventually more powerful theories to be built. If all you are doing is proving everything reachable by construction, you aren't really learning, as this process won't be drawn to the more compressed expressions by necessity but only as a "please" from the humans.
Sure, but we are approaching a time at which we will have to answer a "why" here. I.e. anything beyond checking and validation becomes of mostly artistic value at some point. And there is nothing wrong with that.