People get "flu shots" in drug stores and grocery stores. Why would anyone think that a "Flu Clinic" that injected people with the CURE for the flu wouldn't be similarly popular?
And, since you have to go to some kind of medical clinic to get Tamiflu, what's the difference between getting a prescription for a drug, or getting a shot right there? The shot is actually MORE convenient since you don't have to wait to get a prescription filled.
Let's not forget that there are needle-less injection systems that really work. Individually pre-packaged doses of 'cides also might be something that could be done in the future so that people could self-administer.