What is the point of NPR and PBS?
To provide decent-quality news and programming to all Americans.
Why do you think that the taxpayer and "marketplace of ideas" is a garbage position?
I for one won't say it is garbage, but I will say that NPR in particular, and government-run networks (notably BBC) produce excellent news and programming, that usually fills gaps that are not necessarily profitable.
For example, commercial kids' programming has deteriorated largely into 30-minute advertisements for toys. (This isn't and wasn't always the case, e.g. classic "Looney Tunes" was exceptionally high quality, but is no longer produced.)
Sesame Street would never have developed on commercial TV networks.
We should note that state funding does not mean state
control. Networks like NPR, PBS/CPB, BBC etc receive funding from the state, and face little or no editorial control by the government. In contrast, a propaganda network like RT or People's Daily basically exist to support the regime, spread its ideology, prevent criticism of the regime, delegitimize any opposition, and so forth.
Does not NPR and PBS receive at least some of their funding from the government, i.e. the tax payers?
I believe it's somewhere around 20% for CPB. NPR is around 14%.
For the most part, NPR and PBS are well-funded in most urban areas. Government funding is most critical for stations that cover rural areas, which don't have much commercial programming either via public airwaves.