You want to send potentially sensitive health updates over SMS? Or just a link, that then opens the web-app anyways? Definitely sounds like bad UX.
Push notifications are subject to the same HIPPA rules as sms. You can’t included sensitive info.
Spammy notification requests should be mitigated with better browser guards for requesting them. The fact that there is spam does not somehow invalidate that there are legitimate use cases people want them for.
This includes what vaccine I received, the lot number of its production, plus information from v-safe[1] and reminders to check in telling the government how I feel on days after the vaccine.