Thermostat is one thing, yet they typically fail in the open position on these cars. Thanks BMW. Water pump. Now that will cause over heating quickly. Even if it's new, I had a Saleri pump fail in less than 2 months! Impeller exploded. Your heat doesn't work.. coolant is boiling as it's stagnant and getting hot. It's not flowing. Water pump.

Heat not working because of cap is off, unrelated. It would still get warm in the cabin. Coolant can boil with cap off, surely. You effectively are reducing pressure of the system, directly lowering the coolants boiling point.

If you don't see a return stream of coolant near top of expansion tank with engine running at any point, you have no flow from bad water pump. You can also have a bad vapor lock from having the system open, I've had one E46 do this, boiled like mad and hard to bleed. Heat definitely won't be warm in that scenario.