It can be done by robot, it several new abilities, it acts as a staging post for deep space missions, it lets NASA examine an asteroid close up and repeatedly, and it can be mined, again demonstrating abilities needed for deep space missions and for long-term possibilities for resource acquisition and management.
I think the second reason you listed to be much more compelling than the first. It could be argued that the space race was about ensuring that the US had missile superiority over the USSR, but I'd wager a significant number of the aeronautical engineers and astronauts involved were much more interested in going to the moon because it was there.
... and heck, it gives us a reason to go back to the Moon ... or at least, send more probes, etc, there, which themselves might help give more reason to go back to the Moon.
This is Step A, where Step Z is one of "save the planet from a world-killing asteroid" or "everyone gets a solid gold toilet" (or "a few people get solid gold toilets larger than the existing moon")
The earth doesn't have an unlimited supply of resources. Asteroids are literally loaded with minerals. At some point we are going to deplete the earth and will have to start looking to the skies to get what we need.
The Earth has far more minerals than the asteroid belt combined (its total mass is ~4% that of the Moon[1]). They're just buried under a lot of less interesting minerals. And there's some heat and pressure involved.
I'm not convinced that asteroid mining would be more productive than, say, tapping into rift zones, but it's an interesting problem. Might be useful to solve.
What are you on about? Limitless energy is rained down on us by the sun, and matter itself is conserved at the elemental level (barring nuclear processes, which are minimal on our planet). What do you suppose we are running out of?
Easily accessible resources. Yes, in principle everything can be converted into everything else if we pour enough electricity into particle accelerators, but in practice we have nowhere near enough power or time for that. Not until you get the nanotech working, at least.
So until we'll be able in practice to make everything out of electricity, dust and the power of will, we need various resources for our survival and growth.
Also, why not?