The challenge here is royalties. You have to pay people royalties for broadcasting their work. You could take the position that publicly funded media should be in the public domain, but then it's going to cost considerably more because you'll have to compensate people for the royalties they won't receive.
> I might be wrong in this particular case though — maybe the public funding was always indirect, through the broadcasters?
Yes, Sesame Street wasn't commissioned by PBS, it was originated by the Children's Television Workshop which was funded via grants (some from the state, but mainly by non-profits like the Carnegie Corporation). And originally it aired on the precursor to PBS (National Educational Television), which was funded by the Ford Foundation.