This is a political argument in the economics profession. Beyond the obvious points that math/data can be used to obscure weak arguments and that idealogical binders are bad for clear thought, Romer is simply attacking his political opponents. Oddly going after George Stigler - a Nobel winner who died 25 years ago who Robert Solow said "was never an ideologue."
Neither Kay nor Romer are the clearest writers but I think Romer was clearly attacking Stigler (as well as Lucas and others at Chicago) and Kay was just repeating what Romer said. See this:
I don't have access to Stigler's 1955 paper. It would be interesting to know what he was talking about but putting him in opposition to Feynman seems pretty harsh. It makes it sound like Stigler is a proponent of idealogical thinking and that's how he practiced economics.
