That is how people think about high oil prices.
If oil was to go to zero people would stop pumping it and burning it (for that to happen the alternatives have to be cheaper/better). That is what will fix climate change in the US.
As for the bigger picture — yes, higher prices for oil might spur extraction in regions outside of the middle east, but that's a local only viewpoint. Globally, higher oil prices reduce consumption and make green alternatives more attractive on net.
When oil is high, GDP takes a hit. Less is happening.
1. We stop using oil because we have better, cheaper alternatives, as you already said. Alternatives are cheaper if they become cheaper, but also if oil becomes more expensive. Higher oil prices may stimulate some oil exploration in the short term in the places that have oil. Everywhere else it's going to cause a scramble to renewable alternatives.
2. We stop using oil because we have technologically regressed to the Middle Ages
I prefer option 1.