I get that mining asteroids seems like a very expensive task. I understand that it's expensive to get to and from them and takes lots of propellant. But I just thought of a a way to make it way cheaper, eventually. Could someone check my layman assumptions, please? :)
Asteroids zip past earth pretty frequently. Parking an asteroid that goes by is probably really hard because of the speed difference. When we first start up and deal with already known asteroids (and have lots of time to slow it down) it should be a lot easier to first get an asteroid in a circular orbit between Earth and Mars and then slowly bring it towards Earth until it is captured.
Once it's in a relatively round orbit around Earth it's safer and less expensive to park it around the Moon than Earth. Mining of an asteroid around the Moon is still a big task but it's a lot more manageable than doing so from Earth. This is because as a climber goes from Earth to the anchor it accelerates and has to break in order not to be slingshot out. And climbing is also pretty expensive in the opposite way - it has to fight that outward energy to get back down. This acceleration effect is much smaller on the moon with it having about 1/6 the gravity.
So parking the asteroid around the Moon is clearly more economical and safer. Getting the mined material back to Earth is a separate matter but space elevators help here in a in a surprising way. First off, going back to Earth is easier from Moon orbit than it is from the Moon itself - less energy spent. The energy that is expended by slingshotting a carrier back to Earth will slow the asteroid down and cause it to get closer to the Moon. But this can be offset by capturing carriers that are coming from Earth. And if properly timed this procedure should be both safe and cheap.
The same exact slingshot and capture effect can be used to park carriers by an anchor in Earth orbit. This is a more daunting task but is way more safe and cheaper than parking an asteroid in Earth orbit - the weight of a carrier will be a lot less than the weight of a mined asteroid.
This also helps solve the problem of getting the material down to Earth. First the carriers are significantly slowed down by the capture process. From that point on a combination of shuttle and Starship-like controlled burn technologies can be used to land a carrier down to Earth. Scramjet technology can be used to forego the wait of the oxygen needed by the carrier to slow itself down as it burns propellant.
As carriers enter Earth's atmosphere they engage scramjets and vector the thrust to slow down in a more horizontal manner, thus eliminating the need for heat shields that need refurbishing after each landing. Once at speeds too slow for scramjets to work the same engines should be able to operate in jet mode to slow for landing approach. As it slows down the (now) jet will gradually change the angle of its wings and land at airplane speeds in a very tried and true manner.
The fuel needed by the carrier can be sent up to the Earth anchor in light and soft packages (in slightly cooled gaseous form, with softness and large surface area helping to slow them down at the anchor). It will then naturally liquify and be ready for use by the carriers when they go back down to Earth.
From Earth surface, railguns can be used to quickly get carriers to M4ish at which point scramjets would kick in and get it to low earth orbit. There, carriers can attach to the elevator and gain more velocity automatically. Thus carriers won't need any rocket engines at all and can be much simpler in design.
And the energy needed to park an asteroid in Moon orbit can also be reduced by a Moon space elevator which would (in a synchronized manner) use the energy acquired from the capture to accelerate a caturer to Sun orbit to go after yet another asteroid.
Of course it would still be very expensive to get an asteroid slow enough to get to a Moon elevator. Even with "quickly anchor and slow down an asteroid with counter momentum, mid flight" technology. So we'll need propellant in Sun orbit. Some to get it to a "quick capture orbit" from deep space and some to further slow it down further after Sun orbit capture. But could we mine water on the Moon and use solar energy to separate it into oxygen and hydrogen? And by the time we start running out of water on the Moon we should have fusion rockets, right? :)