I don't think this works because you're basing assumptions on the 21st century practice of dispensing medicine to a patient for home use, with no way of monitoring compliance. In the 24th century, you might trust a patient to self administer a drug, but for sure you are going to have a computerized way to monitor when and how much is injected. Your hypo is going to tell your doctor if you miss a dose or take too much, and a nurse is going to immediately follow up to tell you to take your injection. And if you still don't administer it, they'll bug you more. And that's assuming they wouldn't just implant a tiny slow-release dispenser in your abdomen to begin with. There should be no excuse for medication compliance falling through the cracks by this point in time.
Deleted scene that turns up in the novelisation actually has Beverly put a borderline invisible implant into Zefram Cochrane to moderate his BiPolar funnily enough. Can’t remember if she even had his consent, but it was done as a parting gift from the future he helped create/save.