Based on your approach, LACs with similar stats would do the same as research universities, since the best proxy is stats. But if you look at the Niche poll scores, the LACs consistently do better. You may think this comes down to the vagaries of the poll questions, but why would it so consistently favor the LACs? All respondents are answering the same questions.
I don’t think all institutions are created equal (as long as they have equal stats). It could just be that some research universities put more emphasis on research and graduate study to the detriment of undergraduate study, and that is what the data shows.