My scenario is several orders of magnitude more common than yours. It doesn’t mean you’re wrong; miracles do happen. It must be nice to have so much blind faith in technological progress and human exceptionalism, but I’m going to stay with my ecology and statistics.
It's several orders of magnitude more common in species which have orders of magnitude less ability to impact their environment and change the way they live (and to humans in the past when we were no different).
It is not blind faith to say that we should expect that the different dynamics could be different when the rules of the game are different.
Again, if the foxes see the rabbit population declining and can start producing rabbit substitute before they start starving, of course we would expect a different dynamic to be possible.
By that logic we shouldn't be applying arguments like "It is quite normal in ecology to see crashes and recovery in the event of overpopulation".