Isn't this a contradiction? Colo[u]r being those frequencies of light visible to humans...

All of these arguments depend on the definition of "color", and until we all agree on one, there will never be agreement on how many "colors" are in a rainbow.

isn't the vast majority of the sun's radiation thermal, i.e. blackbody

Well, obviously, essentially all of the sun's radiation is blackbody, in that none of it is reflected or transmitted. :-)

blackbody, i.e. continuous in the frequency domain

While blackbody radiation may be continuous theoretically, that doesn't mean that thermal radiators like the sun have continuous spectra in reality.

Brass tacks: Is the solar spectrum continuous or quantified? If the physics PhD says it's continuous, I'll believe him.

Yes, there are spectral lines ... but this isn't what the majority of radiation emitted from the sun is.. is it?

Is there any reason to suppose that the "smoothness" of the solar spectrum is qualitatively different from one part to another? No; In fact, it looks like the roughness is merely proportional to the intensity, across the spectrum. It also appears that whatever "fuzzing out" is happening, it is not enough to completely smooth out the spectrum. Far from it!