I do "Do Not Disturb" 24 hours a day, 7 days a week. No calls, no texts, no notifications, no nothing. Yes, Apple has indeed built it out recently. At least on iOS, you can add exceptions if you want, like your favorites, or your contact list, or even specific people. You can also put it on a schedule if you think 24/7 is too extreme.
Honestly, I don't worry about the "urgent call from the doctor" scenario, and I have a family. If it's important, whoever is calling will leave a voicemail. For 99% of the calls I get, it's not even remotely important. I don't structure my life around outlier scenarios. Not having a phone constantly dinging and buzzing for my attention is worth missing the extremely rare emergency.