The "unlimited" is mostly branding, like gmail's unlimited email storage. In practice, professionals like myself simply don't have the time to bug you at all hours of the day. I have a dayjob and maybe an hour I could devote to math every weekday. So worst case you'd receive 1 question a day...usually far less. Moreover, students tend to ask the same questions, so after the initial phase you can simply redirect most queries to preexisting URL. Also, you as a professor don't have to chime in for everything - have a graduate student preprocess the majority of the questions and you attack the harder 10%.
Its definitely feasible. Chegg does something similar - what's missing is the quality and rankings
