I don't even need a cellular watch. Just having the phone inconveniently nearby (e.g., in a backpack, or on the bathroom counter instead of a nightstand) adds enough friction to eliminate mindless phone use.
I don't get it. This functionality is so basic that it's always been part of the phone itself. What's the watch doing?
What is the watch adding? Were you having trouble distinguishing calls from notifications before?
If I get a message in the middle of a meeting, I can figure out if it's just my mechanic saying my car is ready to be picked up. Or if it's a severe medical emergency. All this without bothering anyone else with audible alerts or getting distracted by any number of things on my phone.
If you have your own system, that's fine. I'm not saying this solution is for everyone. All I'm saying is that it's a useful solution for at least one person in this world.
The solution might not make sense to you, because you apparently don't have the issue. For some people, a phone making a sound in their pocket, ANY sound, or vibrating, or just merely existing, is an invitation to do some of that sweet sweet scrolling that releases the good chemicals.
A smartwatch (like the Apple Watch) is, at this moment, just smart enough to allow you to interact with simple yet "smart" (i.e. not phone/SMS) apps, like WhatsApp, maps, music, payments, tasks, exercise, but inconvenient enough so that procrastinating on your watch just isn't a thing.
Before: Every time a message arrives the phone makes that noise, and a little lamp is lit in my brain which stays lit until I pick up and unlock the phone and look at the message, answer, etc.
After: The phone is in silent mode all the time. When a message arrives I get a little tickle on my wrist - I just glance down and see who it's from and the lamp in my brain has nothing to do anymore. I even sometimes forget to pull out the phone and answer the message later.
It's easier to leave the phone out of reach, I feel less need to have it on me. It's great having a phone which never makes a sound. If a call comes in I see it on the watch and can reject the call from my wrist and optionally send an SMS instead. And podcasts are not interrupted by annoying pings.
It's a Garmin, so only have to charge it once every 10 days or so, and there's not much to do on it.
I also use greyscale mode on phone, can set a keyboard shortcut. Some apps are also just far too colourful - e.g. Duolingo - and look better in greyscale I think.
Phone can be a productivity machine too though with a Bluetooth keyboard, phone stand. Can have full desktop environment with a keyboard and mouse and VNC client.