As it turned out, that is almost exactly what happened. The bubble of high prices made it economically viable for other rare earth mines to open, and they did. The mothballed Mountain Pass mine came back online in 2012, ramping up to full production in 2015, and in May 2011 the Australian miner Lynas Corporation started production at its Mount Weld rare earth mine. Since then, Lynas has built a refining facility in Malaysia to process rare earth products.
So long as every technology company wants to wait a couple years before making product again, everything will work out fine for capitalism.
I personally would be glad if China did put more restriction on rare-earth metals exports due to this whole trade dispute. It would both stop their reckless mining of it, while bringing prices sooner to what they'd eventually level out at as China becomes more developed, their wages increase, and regulations become more enforced. Yeah, it would cause a shakeup, but it's something that would happen down the line in any case and it would eventually level out.
And I'm willing to bet technology companies would still be able to produce in the interim, China isn't the only country that exports rare-earth metals. Electronics would just be a lot more expensive for a time. I honestly believe the market would respond rather quickly, though, and a lot of mines would be reopened.