Novice here, why don't subscriptions lead to hockey sticks? Which business model does?

Hockey sticks come from low friction. Subscription models are just plain tougher -- people have to put in a credit card and pay real money for the service. You just don't get exponential increases in that type of a situation (cue counter-example).

The business models that lead to hockey sticks are anything where the initial use is free to the user. Essentially, how Google, Facebook, and Twitter all started. They can use advertising to monetize that growth, and eventually ancillary paid services (Google Drive storage for example).