If you want to be more inclusive, have a sign like, "Tell your cofounder/babysitter than in an actual emergency they can call or text [number] and we'll come get you."
I could happily live without a phone. As it is, according to the Android "Digital Wellbeing" tool, I've opened / unlocked my phone only 10-20 times per day for the past week. I actually hate my phone (it's a three year-old budget model) and prefer to do everything on my laptop when I can sit down somewhere and focus.
But I have four small children, and my wife would not be pleased if I told her that I was just getting rid of the thing and would become harder to reach, or wouldn't be able to make an emergency phone call.
However, I'm also in the military and have worked in a number of environments where I couldn't have a phone. In those environments, there's always a 24/7 duty number that I can give my wife to call in an emergency, which someone will answer and start a process to find me.
Edited to add: In either Deep Work or Digital Minimalism, Cal Newport says that his wife made him finally get a phone when their first kid was born.
Having lived in a really unsafe enviroment, there's too many variables that people who live in safe places take for granted, Have you ever been followed by a car? or have you had a family member kidnapped? Calling the right person (Because local LEO wouldn't do enough until an actual crime had taken place, has saved me and my family more than once)
Your kid being followed in a car and kidnapped doesn't change because you are by your phone. If the police ignore you calling about a car following you the cell phone isn't going to help
In your situation I would trade in your cell for a gun. It gives you a fighting chance against kidnappers and I would not let my kids out without armed guards. Whatever protection a cell phone provides parents in the first world those benefits apply less in your situation.
Allergies that weren't discovered in a medical setting aren't on file, I've had that happen to me
From a recent break in near my house, the only reason we could act timely, was because we called the owner, confirmed the house was supposed to be empty at that time, and then coordinated
Having dabbled with armed guards, unless you're high profile enough to keep a larger force, it tends to just put a bigger target on your back
Kid's medical file is on a computer in the pediatrician's office. Kid gets taken to an ER at a hospital with which the pediatrician is not in any way affiliated. How does the emergency doctor quickly get the kid's records? Can the emergency doctor even find out what pediatrician the kid goes to unless a parent tells them?
Personally, I'm waiting on a call about an adoption, and I can't ever leave my phone until that goes through. Even before cell phones, parents would leave a number for the babysitter for where they'd be. Yes, kids were raised without phones before, but I'm sure it wasn't a better situation when parents got home and found a note that their kid had to go to the hospital or something. People may say this is fear-mongering, but I guess they don't have kids with health issues.
If you don't want to be distracted, there are features like Screen Time that can lock you out of every app except phone and messages if you want.
Also, not a daycare center, but a babysitter. For many of the babysitters we get, we're meeting them for the first time when they show up at our door; that's just reality these days. And it's not uncommon for them to text us while we're out, though thankfully no emergencies so far.
Its like saying "just use a pay phone, like we used to do". You could, except there isn't a single payphone in my country.