No, if we really want to show we belong we need to show that we have a following that won't mind being inconvenienced a little to support the programs.
Alabama (I'll let you make the call on whether they qualify as big time and at what point in history they earned that qualification) played every important home game in Birmingham, not Tuscaloosa until the mid 1970's and approximately one game a year for roughly the subsequent dozen years. Keep in mind that Bear Bryant won a number of national titles at Alabama, including one as a player.
