He's actually doing something really deep here. There's an old medieval concept concerning kings, relating
the body politic and the body natural. The king embodies two sides: political power, which is greater than any man; and humanity, which tempers the coldness of the law. Both bodies are united in the person of the king, who literally personifies the health of the nation. This mirrors Christian theology where Christ was simultaneously 100% God and 100% man.
For all the pros and cons of monarchy, one of the obvious things it does is give a visual representation of the national state. And when the king gets sick, it implies the nation itself is sick. This association shows up in a bunch of different cultures, not just English Middle Ages. Depending on which culture and time you're in, it is taken more literally or more metaphorically, but it's still a powerful symbol: an ailing monarch reduces the confidence in your nation.
The modern emphasis on a person's mind over their body is relatively new to the collective subconscious. Our "mental software" hasn't fully upgraded, and we sill have these kind of associations to the people we look to as leaders. The medieval political theology was just a written-out explanation of what people naturally gravitated to, and when we lack alternative "intellectual" institutions that rationalize things otherwise, we fall back on these tendencies.
TL;DR Which is a long-winded way of saying, Trump had to prove he's healthy in order to make the nation itself feel healthy. The claims that he was sick was a
political attack on his actions, not just on his person.
I don't think Trump is a medieval theological scholar, of course. But he intuitively understands the need to show
personal health in order to restore
political trust. Especially at a time when viral sickness is rampant (Kung Flu), and the political order has collapsed in cities (CHAZ). That's a double-dip into symbolism, and he addressed it head on.
