You have two loops in your BGR.
Opamp- to M3_gate to M3_drain to Opamp-
And
Opamp+ to M4_gate to M4_drain to Opamp+
One of them is a positive feedback loop and the other is a negative feedback loop.
You want your negative feedback loop to have more gain than the positive feedback loop and that happens by design in the above schematic.
Now if you flip the opamp inputs, then you have got more gain in the positive feedback loop. and hence your BGR is dead.
Also, you need to check the stability in-situ. That is, with the opamp as part of the loop.
Your negative feedback loop mentioned above needs to be stable. Not just the opamp alone.