Larger pixels do generally help at the individual pixel level, so your intuition is reasonable. The key is that sensor performance is not determined by pixel size alone.

Newer cameras can achieve better high-ISO results through many incremental improvements, including better microlenses, gapless microlens designs, cleaner readout electronics, lower-noise amplification, and on-chip noise reduction. These improvements reduce losses and lower noise introduced after the light is captured.

So while shrinking pixels can hurt per-pixel noise performance, advances in sensor and electronics design can more than offset that. Also, when images are compared at the same final output size rather than zoomed in pixel-for-pixel, the difference is often much smaller than expected.

Diffraction is different: that is a physical limit, and smaller pixels make it visible sooner. As pixel pitch decreases, diffraction softening becomes noticeable at wider apertures than on lower-resolution sensors. In other words, yes, smaller pixels move the diffraction limit closer.

So the short version is: smaller pixels alone are not the whole story; newer sensor technology can improve usable ISO, while diffraction remains governed by physics.