The client initiating the 0RTT provides a pre-shared key, thus revealing to the server that they're not a newcomer. I don't know exactly how many bits of that PSK could be used by the server to identify specific clients. For QUIC I think it's a 15-bit identifier. Browsers will need to clear the PSK (and so remove the 0-RTT) when they clear cookies or in a "private browsing" mode.
I mean, if I want to get weather data from let's say NOAA, so a simple GET / HTTP/2, why would I want to send any PSK? Let the server send the response and the Server Cert and the client can decide whether to trust the reply or not.
CloudFlare only "allows" 0RTT for GETs, for example. Is that different, or they also need the PSK?
For the Internet of Things it's also envisioned that some devices might know a PSK at the outset to use TLS rather than some custom protocol to secure their traffic. Maybe your lightbulb controller knows a PSK for the lightbulbs baked in at the factory. But it's not expected that web browsers will care about this case.
