>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.