I'm not sure I would worry about him safety wise. But I would wonder why he is so obstinate about things that make life simpler for those around him.
We were out of town last week and my husband accidentally forgot his phone at my parents one day we were out and about. It didn't create any major problems, but a lot of small inconveniences could have been avoided if he had had it with him. Exactly like you mention, going one way with one child while I went another with the other child. We were shopping. But had no idea if he'd left the store or where he had gone, or if he was done and gone back to the car to wait. He was in the same boat, not knowing where I was.
Of course, some of that could have been avoided if we weren't accustomed to being able to go separate ways and calling each other to meet back up. We didn't designate in advance where to meet. A habit we've let go since we both have phones, but used to use when we didn't.
If he is taking the only means of transporting the kids and doesn't have the kids with him, then he should be accessible (communication wise).
Maybe rather than explain the logic of it, since he is arguing logic back to you (we got by fine before...), a better approach might be to explain how it makes you feel, and that you would feel better, more calm, more relaxed, if he had it with him so you could reach him easily if you needed to.
Because, honestly, it sounds like a bit of rebellion on his part. The best way to get him to come around isn't going to be to show him how wrong he is, if he is rebelling. It will be to explain how much you would appreciate him doing it FOR YOU. And understand that it WILL be "for you"... not because he agrees with any logical benefits you are pointing out.
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ETA
I also wonder if he is afraid of becoming "one of those men" who is constantly barraged by their wives with messages and calls. My husband always takes his phone (unless he just plain forgets to grab it, which happens), but I rarely call him. He gets more texts from guys at work than from me, generally. He uses his phone to call ME more often than not.
But, he has friends (same age) who almost never use their phones. They also are the group that avoids the newer technology and has no desire for "smartphone technology." He is golfing today with his friend who always has his phone, but always has it turned off and almost NEVER uses it. I know this from his wife. He just doesn't like using it. It isn't specific to her. My own husband gets annoyed sometimes by people leaving voicemails. He has a difficult time navigating through to make the notification disappear.
So, another thought, is it possible your husband is not comfortable with the technology and doesn't want to be forced to use it out of that whole "manliness" thing, where he feels he has to be competent at everything? Or something like that?