How does an op-amp become stable in this circuit?
You've got the explanation right, it's just that you don't understand the simulator. It has found a solution and presented it. What it has not done is a stability analysis, and in this case the circuit is not stable. The simulator has not varied the current, but rather found a set of values that "work" for a single, exact set of values.

Try this: replace your 1 mA current source with an AC current source. Let's say +/- 10 uA with a 1 mA DC offset. Perform a time-domain (transient) analysis on the circuit and watch what happens in each circuit as soon as the current varies from exactly 1 mA.