I still work with technicians and other staff who don't have cellphones. (Or if they do, only turn them on when they're expecting or making a call.) Modern theatre works just fine in a landline-only world: it's less convenient and you don't get instant responses (where are you, why aren't you here, change of plans: we need you to swing by the party rental place when you're finished with the LX rental, etc.), but you can correct for most of this through planning.
F'rinstance, it's true that, without a cellphone, you can't reach someone when they're late. So instead you make it their responsibility to reach you. Here: this is a business card with the Running Number on it. (The Running Number = a landline somewhere in the theatre on a desk which is staffed for at least the first half of the show: box office, stage door, house manager's office, etc.) As soon as you know that Something Has Gone Wrong (the subway breaks down, you get stuck in traffic, you're sick, whatever), go find a payphone (yes, they still exist) and call this number. Crucially, you should do this even if it involves a detour: I'd rather you be five more minutes late (with some notice of when to expect your arrival) than for you to be "only" 30 minutes late but have no clue when you'll be in the door.
There. It's not a perfect solution, but we can run a theatre under these circumstances.
Conversely, I'm not sure I'd consider it baseline safe to run a modern automated show without headsets. Even without automated elements, I'd still take the headset over a cellphone.