It would be a lot like asking the kids who attended school before mobile phone ubiquity to use morse code over telegraph lines instead of touch-tone dialing on the public phones or the phone in the office.
If you want to force kids to live in the past, do it in history class. Otherwise, adapt your policies and curriculum for the times in which we all live.
Or it would be like expecting a kid able to send a morse code telegram to write a letter and have it delivered by the postal system, when the morse key and telegraph line is right there.
Or it would be like expecting a kid who knows how to write a letter and use the postal addressing system to inscribe and fire a cuneiform tablet then pay a random itinerant to carry it to the next town, when there's a post office right there.
It's unnecessarily forcing someone to use the previous generation of communications technology.
Also, as far as I know, there wasn't ever a time when people knew how to send telegrams, but didn't know how to write letters. Indeed most people didn't know how to send telegrams with a telegraph key at all, they'd just write up their message and take it to an office to be keyed by an operator. A process which is largely identical to posting a letter.
But if you know the number, you can use your friend's phone in a few seconds.
And yes, the secretary/front desk/whatever can call me on behalf of my child, but that seems pretty ridiculous, too when everyone already owns a cell phone.
[1] My kid is two years old, so this is obviously academic for me at this point.
Of course, and you probably couldn't even find one in the first place.
But you could borrow a friend's phone, or ask a teacher to use a school phone, or even just ask someone in a random store if you're desperate. Just having access to that number without a charged mobile takes you from helpless to having many options, and it takes minimal effort. (Hell, write it down if you have to.)
It makes no sense to expect students to use a payphone in this day and age. How many schools even have working payphones now? I bet most had them removed or disconnected because no one uses payphones anymore.