Engine failure

I think your mechanic was right. RPM has nothing to do with whether the
engine is getting gas. It will windmill at 800-900 RPM with the engine
off. I have seen this problem before. It is not extremely rare. An idle
check on runup should discover it, unless you finally managed to loosen
the throttle just enough on short final for it to finally stop working.

The reason I do not think it was carb ice is that the weather was below
freezing, which means that the air was probably relatively dry and it
may even have been too cold for carb ice to form (the air would have to
have liquid water precipitate out of it to cause carb ice, meaning that
the air would have to be warmed in the carburetor, not cooled, and
somehow also exceed 100% humidity in the process).

This is one reason many instructors are so dead set against their
students getting low on final. You can run out of options real fast,
and a disproportionate number of engine failures happen there.

Engines almost always fail just when you change something: throttle,
carb heat, mixture, prop, etc. There can be a lot wrong with an engine
and it will keep running if you never change anything, but a
configuration change will often be the last straw before she quits.
Before making any adjustments to an engine it is good practice to scan
the area for emergency landing fields.