It wouldn't bother me if they were required to transmit this information via radio, though of course for ground vehicles one can simply visually look at the plate number (something impossible to do for an erratically flying aircraft, at least without a killer zoom lens and tripod).
And then, for DOT hazmat licensing, which is perhaps more comparable to a private jet in terms of risk, the registration requirements are quite stringent, and including prominent display of DOT hazmat labels, as well as licensing information which is of course public.
It seems to me that private jets just became far more opaque than semi trucks, relative to their transport mode.
I'm unclear what jet aviation risk we are equating to hazmat ground transport. It's not raining aluminum very often, and it's a news frenzy when it does happen.
Every time I read "private jets should X" I can only ever boil it down to envy or "because I have the right to know", which I personally don't agree with. I seldom see things like "transporting that organ to its transplant patient via light jet is so carbon intensive -- why not use passenger aircraft cargo instead?" or other more nuanced takes.
For some, I imagine the opacity is worth the frightful cost of private aviation. I don't know what it is like to have millions of fans trying to figure out where I am at every waking moment, and the very idea is horrifying. I don't think any valid purpose is served by knowing where <star du jour>'s jet takes off from or lands at, and I don't think anyone is honestly looking skyward to FL410 and worrying about an aircraft part landing on them.
From other conversations about this topic, I thought it was interesting that the congresscritter who advanced the legislation is from the Bentonville AR region -- and crafty investors have tracked Walton jets to figure out where the high-level Walmart business meetings are happening, then making inferences about upcoming stock-moving deals. I thought that was fascinating. Humans are crafty, and humans with means who want privacy will find it, despite any number of laws passed. It all feels like wasted whack-a-mole effort to me.