>Mathematicians can have as much fun as they want while working on math problems, they have the skills and the brains, why shouldn't they enjoy their work?

Obviously. In my ideal world even the janitors at the same institutions would have a blast, it's just not usually the end-goal reason humans are hired.

>With math you never know up front what 'real improvements in engineering and other life improving fields' will require some key bit of it that someone interested in pure math came up with a decade or two (or 10) before we found an application.

Right, but would it have made a difference for the application layer if that key bit of "pure math" had been discovered by a human (that had a great time) or a machine? That's my point about games being a bad analogy.

That depends. Many of these things can only be applied safely once they're understood.

For instance, almost all of cryptography is applied math. If your opponent understands the math better than you do you might end up with your secrets compromised.

Math is about as 'pure' a science as there is, the touchstone of applicability is missing the point entirely, it is first and foremost about understanding things at one of the most fundamental levels (even more fundamental than physics) that there is.

Nobody cares all that much about the discovery (except that we name things for the person that discovers it, it's sort of analogous to the discovery of a new piece of land). And given that the pay usually sucks this is a nice thing to be able to do. What mathematicians usually do really care about is their ability to understand the discovery and that this discovery has the proper foundations (other pieces of math that are also understood). It doesn't really matter what the answer is if you don't actually fully understand the question or the foundation the answer builds on.