You are correct, except I don't like the way you said it. I think it is more accurate to say that full frame has lower noise levels because they tend to be able to achieve the equivalent of lower ISO.
In order to get equivalent ISO, you need to multiply it by the crop factor squared. (Squared because the sensor area is two dimensions.) So for MFT 2x crop factor, that means 4x. Which means a MFT sensor at 800 ISO has the same theoretical noise level across the frame as a FF sensor at 200 ISO.
This is another area where the way we describe crop factor does a huge disservice to practical application. By pretending that smaller sensors are just cropped versions of larger sensors, we have to do a pile of stupid math in order to find equivalence. And it causes no end of pointless internet arguments.
https://petapixel.com/2017/03/29/dispelling-myths-around-dep...
25MP (or any resolution) full frame sensor with a (say) 1.8 lens will gather more photos than 10MP (or any resolution) APC sensor with a 1.8 lens.
Thinking about the individual pixels of a sensor is tempting but causes an unfortunate chain of erroneous assumptions.
It is true that (a) larger sensors have had an historical efficiency advantage, but it's progressively diminishing, and (b) larger sensors tend to be capable of lower area-equivalent ISOs. If you could build an MFT sensor capable of ISO 25, and used comparable glass, the full frame advantage would be eliminated.
Noise only matters in the context of output. If you’re printing—again just an example—a full magazine spread, the noise level in the final result isn’t fundamentally affected by whether the sensor was 30 or 50 megapixels; FF or MFT. Either way it is ultimately scaled to the print size and print resolution. Nobody except the photographer cares what the noise looked like at 100% pixel zoom. All that matters is the relative appearance of noise in the final product.