Do these other fields also study humans in controlled experiments?

I think it has to do with the sample to staff ratio. It's not enough to observe human subjects. You have to actively prevent them from going off the rails. It doesn't scale well when you increase the sample size. I guess we could replicate a similar experiment n-times and then do a meta study, but it's not ideal either.

How would you tackle the logistics of scaling up the above experiment?

Yes, the most common example would be clinical trials for drugs and other medical treatments- often have thousands of patients (with recruitment being the limiting factor). There are tons of ways that studies can go wrong, for example when patients don't take the treatment and lie (this is common) or have other lifestyle factors that influence the results, which can't be easily smoothed out with slightly larger N.

I don't know how to fix the nutritionist studies- I'm still pretty skeptical that you could ever control enough variables to make any sort of conclusion around things with tiny effect sizes. This isn't like nutritional diseases we've seen in the past, for example if you look at a disease like pellagra (not getting enough niacin), literally tens of thousands of people died over a few years (beri beri, rickets, scurvy are three other examples; these discoveries were tightly coupled to the discovery of essential nutrients, now called vitamins).