Going "full on jazz" would entail improvisation, which is something the band members have said they suck at.

I think Radiohead have made one song in their whole career that could be characterized as a jazz piece, Knives Out. It seems to be an improvised piece around a set of simple chord changes from one of their earlier songs. And after making it, they spent a whole year trying to re-record and perfect it, until finally coming back to the "imperfect" original and releasing that. Jazz does not come naturally to them, even though they can create approximations by using guest jazz musicians (TNA, LIAGH). The drums of Pyramid Song, the basslines of Bloom and Airbag, the vocals of Jigsaw and There There's verses, are very jazz... but the whole song is never jazz.