A bigger sensor alone gives better low light/high shutter speed performance.
The truth is that the advantage only comes when the larger sensor is used with about the same ISO speed, which in turn means using lenses offering about the same f-stop choices, which with the longer focal length needed to get the same FOV means larger effective aperture diameters, and so generally bigger lenses. When instead a smaller format can be used with a lens of equally large aperture diameter, and so with a lower minimum f-stop, then you get roughly equal lens size and the ”extra lens speed" of the smaller format and the "extra sensor speed" of the larger format roughly balance out:
”big glass" is the ultimate source of good low light performance (and low DOF), not just "big silicon".