I don’t see why you think that. I could. I would miss the milk and meat, but the rest wouldn’t even be that hard for me.
As to the larger question… I have a friend who is deathly allergic to nightshades. He says he reads, “gluten free” as “TOXIC!!!” because a large fraction of gluten-free foods replace some of the wheat flour with potato starch.
People are actually somewhat metabolically diverse. Quite a lot of people have trouble digesting FODMAPs, and a lot of the people who feel healthier when they go gluten-free are actually benefitting from reducing their consumption of FODMAPs. Some people are actually gluten intolerant. Lots of people have trouble with dairy, in one way or another. There are all sorts of weird food intolerances out there, too – I know a woman who had all sorts of health problems until she worked out that she has sensitivities to chicken and corn. Including foods made with corn syrup or chicken eggs. (She can eat duck eggs with no problem, also turkey.)
It’s unlikely that either milk or nightshades are a net health negative for most people. They are, after all, dietary mainstays of some parts of the world. Might they be for Brady? For you, gentle reader? It’s entirely possible.
As others have said, there’s nothing unhealthy about the diet Brady appears to be on, it’s just limited. I wouldn’t go recommending it to everyone. But if you have arthritis and wonder if nightshades or dairy might be contributing, try avoiding nightshades or dairy for a month and see what happens. Better yet, spend a month keeping a diary of what hurts when. THEN go off nightshades or dairy for a month and continue keeping that diary. And then compare, with actual dates and times and degrees of pain, so you don’t just blurrily feel like “heck, maybe it helped”.
Then, if it looks like maybe it helped, consider if it’s worth it to you. More likely, you’ll find it didn’t make any difference. Or you may actually feel less good. But hey, it’s not all that hard to mess around and test it.
Simpler, keep a food diary and a symptom diary. My cousin discovered she’s sensitive to wheat that way. (Her doctor recommended it, as her complaints were common among those who are sensitive to wheat. But you could do it on your own.)
And if you love nightshades, just keep eating them. If you had a serious nightshade problem you would know by now. Like my friend.