You are correct - the motor does run out of "grunt" at 12,000' - because the MP isn't available.
Think of it this way. Take an engine to 6,000' MSL, read the MP gauge before you start it. Say it reads 24". Now we start it, and run it at wide-open throttle. It shows 24" (we will ignore losses through air filters, ect). We measure the horsepower - say we get 140 HP at 2700 RPM.
Now we take the same engine to sea level. Now the MP gauge reads 30" before we start it. So we start it and run it at 24" MP and 2700 RPM. We will get less than 140 HP. Why? because the engine is working harder to expel the burned gases against the greater ambient pressure at sea level. On the intake side, the engine doesn't know or care why there is only 24" MP available (throttle plate or ambient pressure), but on the exhaust side, the engine at sea level feels like there is a "throttle" on the exhaust manifold.
Of course, if we open the throttle fully on the sea-level engine, we will get more than 140 HP, because there is 30" MP available at sea level, whereas there is only 24" at 6,000' MSL.