Nowadays, yes — with a couple notable exceptions, discussed below.
In the past, I immediately accepted / allowed notifications for apps like task managers and calendar — because I figured I needed alerts about important deadlines and appointments so I wouldn't miss them! But now even on those I allow badges but not notifications, because I realized that needing them meant I wasn't doing what I needed to do: work in such a way that I know the deadlines by heart and I fulfill them early, and work from my calendar (also knowing it more or less by heart, having a good sense of time, and disciplining myself to stay on track / on schedule and to respect time agreements I made with others and/or myself). Practicing and developing the discipline was what I needed, and the notifications were a crutch and further atrophied my self-management and executive function. (And too many notifications kept me in a tizzy and interfered with my work anyway — a vicious cycle!)
Now the exceptions I mentioned. In my first response I said that I only want to get notifications from Messages. That wasn't entirely accurate. If I set a timer (say, for a work period) or a fleeting reminder (say, to remind me that my laundry needs to be switched to the dryer in an hour) then I want to hear a notification. That also goes for the one third-party app I downloaded, kept, and use with "notifications": a wake-up alarm that uses haptics only, vibrating to wake me up without disturbing my significant other. The unifying threads of these examples are that they are time-based; notifications are essential to the apps' functionality (e.g., you can't have an alarm without some kind of alert or notification); and I'm still specifically directing the apps to notify me about one-off items. None of these notify me without my say-so, and I never get an unexpected notification from them.
> Have you ever kept an app around without notifications because the pull was strong enough on its own?
Well, if I understand the question correctly, since I default to no notifications, every app still on my phone got kept because it proved its value and non-annoyance / user respect over time.
If instead you're asking whether I immediately decided to keep an app always and forever because it was so valuable, no: apps can and do change, and my needs change, so apps are always on the chopping block and must perpetually provide value. And I regularly review all my apps for that value and delete ruthlessly.
