basketball school more than anything when I was a nonmember kid in the East. Football has supplanted baseball as America's favorite sport over the past 50 years. Coincidentally, BYU became good at football and began winning during this timeframe.
You're not really arguing against the notion that most people are casual, fairweather fans and jump on the bandwagon only when the home team starts winning, are you? For example, I can remember my non-sports-fan in-laws sporting Detroit Tigers baseball caps when they made the World Series years ago. They never before or since mentioned the Tigers or even so much as watched a game until Tiger fever hit Detroit that year. This phenomenon of most people getting on board when the local team wins is undeniable; look at what happens to the sale of team swag when a team starts winning. (The 20% in the general population of us who comprise hard-core spoorts fans, like us CBers, are a little less fickle than that.)
Another example is that Duke has a football team and its followers; however, Duke is still a basketball school, even when the FB team has a good season. If Duke had a 20-year run where the basketball team finished regularly in the bottom of the ACC and the football team made the CFB playoffs 10 of those 20 years, we would begin to call Duke a football school.