I think the problems in nutrition are far, far deeper (we cannot properly control diet in most cases, and certainly not over long timeframes; we cannot track enough people long enough to measure most effects; we cannot trust the measurement i.e. self-report of what is consumed; industry biases are extremely strong; most nutrition effects are likely small and weak and/or interact strongly with genetics, making the sample size requirements larger still).

I'm not sure what you mean by "experimental psychology" though. There are areas like psychophysics that are arguably experimental and have robust findings, and there are some decent-ish studies in clinical psychology too. Here the group sizes are probably actually mostly not too bad.

Areas like social psychology have serious sample size problems, so might benefit, but this field also has serious measurement and reproducibility problems, weak experimental designs, and particularly strong ideological bias among the researchers. I'm not sure larger sample sizes would fix much of the research here.