Impact sites of larger asteroids are also great targets for mining operations. For example there's some speculation about the gigantic impact crater on the far side of the moon containing a gigantic metal lump below the surface
It's not entirely clear to me why you'd want to find asteroids on the moon rather than finding them in space, where the mass wouldn't hardly have to be lifted.
On paper, it would be easier to lift them from the moon than it would be to intercept and slow them down enough to land them on earth. When the asteroid hits the moon it expends huge amounts of energy in the impact, slowing it to the a near-stop as compared to its speed while free in space.
It may be possible to aerobrake asteroids at orbital velocities in earth's atmosphere, but that is a VERY dangerous proposition. It might work for small rocks, but a commercially-relevant chunk of iron would be a very dangerous object to point at earth.
The problem with asteroids is the population dynamics and time required with these heliocentric orbits. Yes, there are asteroids at a lower Delta-V than the surface of the moon. There might be only a handful that are even worth capturing and mining. If this scaled up, however, you would quickly deplete this population of NEOs.
While the Delta-V might be less, it's not less by MUCH because you spend most of your fuel getting to Earth's escape velocity in the first place. The Delta-V benefit is also balanced against a massive time lag involved in sending a robotic spacecraft to capture and boost it to an orbit where we can better access and refine it. Or you send the refinery to the asteroid for larger asteroids. Problem there - you'll basically have to build a new refinery for each asteroid. Even for larger NEOs at higher Delta-Vs, the mass is not all that much. It's also not clear that microgravity will be any better than 1/6th gravity.
There are very strong arguments for Lunar mining. We're talking about ~1 week for transit time from Earth to there. The moon can build vast Earth-like industry. You get to cherry-pick a wide variety of asteroid material types (except for volatiles) without building a new factory for each. Surface transportation isn't particularly hard. Materials to build solar panels are widely available.
Asteroids will be useful at some point, but only after we can reach the frost line. Otherwise, the asteroids we get will be parched and not offer much beyond what's on the moon, or what we can lift from Earth. After we get to the frost line, we will get vast troves of organic material, but this is at tremendous Delta-V cost and outrageously remote and time consuming to access.
A space elevator is really an optimization in reducing fuel costs for upfront investment. Such a thing becomes attractive past a certain volume of launches.
While it’s never necessary, it will make sense at some point.
Without an atmosphere can use coil guns etc to launch projectiles just as easily. You get more cargo capacity for lower investment and can still operate on solar or nuclear power.
If the moon rotated faster a space elevator might be more interesting but it would just be really slow, expensive, and a large risk if it where to break.