However one of the more likely methods of construction (from this distant point anyway) would be a launch of lots of material, tie it all to some asteroid or other large counterweight that has been put in space a good 90,000+km away, and then unspool it all to the surface, which would require a good and cheap heavy launch capacity

(I suppose you could say if we are going to do such an undertaking as build a space elevator, with all the billions it may entail, then launch costs to get it going will probably be the least of our worries, but as far as I see it, every little bit helps!)

Earths gravity is such that you'd need a material for the tether that is stronger than any material that we are currently aware of.

So there is no point in getting any of the mechanisms in place or developed in any detail until there is a material that can withstand the forces in play. Right now, sad but true, an earthbound space elevator is science-fiction.

10.1016/j.actaastro.2012.01.008 (Towards the Artsutanov's dream of the space elevator: The ultimate design of a 35 GPa strong tether thanks to graphene, N.M. Pugno)

For someone that is exploring the materials we will likely use in any attempt at an elevator.

Yes, we can't make the materials today. Like, right now. Or even next month, especially in the quantities needed. But we also can't make the rockets that might make it more accessible. And, more than likely, the first space elevator will be built on the moon, not on earth, which will definitely require heavy lift and would probably be a proving ground for the technology and materials.

So yes, science fiction, but so is pretty much everything else that spaceX wants to do.

They always say the future is just around the corner and in this case they might just be true. Or at least, as you can tell, I hope it to be true.

By the looks of it, the future generations on earth will look at our chase for unobtainium in the same we look at alchemists today.

Nobody's achieved it yet, but plenty of people are working on it.