There are 'secret' launches for the DoD but those usually get announced just a very short time before launch. So it might be a week or so before SpaceX will be like 'next week we are gone launch NRO-balbal and then not provide footage of the second stage'.
However, this was not such a case.
No, it is not possible. Even you you managed to sneak a launch by the 7 billion people on the planet, Russia, China, and the USA all have satellites with sensors built to detect just such launches.
Obviously such locations are constrained by orbital considerations, but also there is FH which may be able to mitigate some of that if money were little/no object. We already know SpaceX has people with clearances that work with classified payloads semi-regularly.
Stealth satellites are also a thing. I'm not sure that I am fully up to date on the state of the art here, or that being such is even possible outside of national military efforts.
Could you elaborate on these sensors you speak of? I know there are ground-based optical tracking systems in use, but the satellite-based surveillance I have heard of is mostly to detect nuclear explosions, not rocket launches themselves.
They just (yesterday) did a publicized launch, is the claim that this is the second stage from Starlink-22?
Edit: It does appear that that is what the news service is saying that the NWS said SpaceX said.
Edit2: Someone on twitter is saying this is the second stage from the 4 March (Starlink-17) launch: https://twitter.com/planet4589/status/1375301028514500615