alt Hacker NewsLeica started doing this a few years ago in response to the first wave of AI images[0]. Other, bigger manufacturers (Nikon, Canon, Sony as well I believe) have also joined, though with less fanfare. Adobe is in the loop.
As someone with a passing interest in infosec and cryptography, I'm sceptical of the long-term viability of this kind of product; it only takes one person successfully extracting a signing key to undermine the entire project.
[0] https://leica-camera.com/en-int/news/partnership-greater-trust-digital-photography-leica-and-content-authenticity-initiative
Yes, you're correct about private keys getting exposed, but it's better than nothing. I suspect though, even after key exposure there may be a way to make new private keys so that compromised keys have a known point when they are compromised, which makes public how much skepticism we should all have about authenticity.
I just think there is a world of difference between "certainty" and "plausibility" when it comes to videos on the internet. Yes, state actors might circumvent it, and skepticism should remain, but there is a world of difference between North Korea trying to convince me of some political scandal, and Pepsi Co trying to convince me that someone I trust loves Pepsi.