They don't conduct the level of research that the AAU values, though they've increased their research expenditures over the last decade.
https://ncsesdata.nsf.gov/profiles/site?method=view&fice=1840
If they wanted AAU membership they could have a spot - the AAU cleared a couple of spots within the last five years. Syracuse resigned their spot and another school was kicked out by a vote of the members - can't recall who that was. Can you?