> pure mathematical research into elliptic curve cryptography based on algebraic geometry and number theory, a solution looking for a problem if there ever was one.

This is ahistorical. Elliptic curves had been around for a long time, but elliptic curve cryptography was applied from its birth in 1985. Also the reasons elliptic curves had been studied for so long (Mordell-Weil Theorem, Falting's Theorem, Sato-Tate, etc) didn't really have much to do with the reason it was proposed, which was that its group probably wasn't vulnerable to the same kinds of attacks as DH.

> When the problem eventually came along, the solution was obvious thanks to pure math.

Also ahistorical. ECC wasn't proposed until almost a decade after DHKE (possibly only that early because it was hot at the time thanks to Lenstra (https://www.math.uwaterloo.ca/~ajmeneze/publications/ecc.pdf)) and it took decades to get in common use. Doesn't seem all that obvious.