FSR3 has been available for almost a month, though?
Yes and no. Does it count as available if only select game developers get access to it? Somewhere (I'm not sure if it's in an FAQ document or a press interview) AMD said about open source code availability that they need to iron out bugs first, basically admitting that FSR3 is beta software until they upload the source code and so far https://github.com/GPUOpen-Effects only contains an FSR2 update from August.