There's no paradox, because your logic is flawed. Bitcoin price won't reach infinity or any unjustifiably high value, it will never consume more electricity than the governments would allow - there are already precedents when governments had to restrict mining because it was straining their energy grid too much.
Bitcoin price doesn't need to be stable, and in fact no currency actually has a stable price - it's all just a degree of volatility. So, again, there's no paradox with "bitcoin stability" and "world stability".
It needs to be stable if you want to use it as a currency. And yes, the USD has a stable value. It may depreciate slowly over time, but the value of the dollar doesn't swing by wildly on a daily basis like bitcoin. The difference between the USD and bitcoin is that the USD is a useful currency and bitcoin isn't.