It is not equivalent. Rare earths are, as you say, not actually that rare, but they are still a finite resource, and the CCP quite publicly discussed that it isn't a good idea to sell their domestic stockpile internationally while a significant amount of their economy runs on it. They raised prices to factor in that future availability might be more important than short-term profit.
The chip ban on the other hand is about R&D and labor, both things that do not diminish over time. Instead, the ban seeks to slow down Chinese advancement in areas relying on those chips, AI in particular. Both measures will lead to short-term issues, long-term lost growth, and mid-term new industries in the respective countries/markets.
> Now many extraction projects are starting to roll around the globe because it has become clear China was willing to use access to them as leverage.
That happened in 2018 too. All the projects at that time broke because China does it cheaper.
The thing that isn't available in most countries isn't the minerals.
> this kind of power is illusory. If you ever use it, you lose it.
But the threat of using it can tie up a significant amount of your adversaries' resources.
My guess is that China will be highly reluctant to restrict exports of manufactured goods going forward. Doing so would directly threaten their own power base, just as the Trump administration's actions are currently taking a sledge hammer to the U.S.'s power base.
Ultimately, this kind of power is illusory. If you ever use it, you lose it.