I wouldn't want you driving me if you kept on fishing a phone out of your pocket to tell you the time.
I wouldn't want you driving me if you kept looking at your wrist, rather than the road, to tell you the time.
This is a very stupid argument.
A few observations:
1) We don't need constant updates on the time.
2) Clocks are everywhere, including cars. That wasn't always the case, some of us are old enough to remember when a clock in a car was a luxury item. But now they're built in to pretty much everything electronic, so watches are less necessary than they were a few decades ago.
3) Some people have an social/emotional attachment to wristwatches, apparently using them as some sort of status broadcast. I don't understand that bit of primate behavior, but the use of wristwatch as ornamentation by some subcultures should be noted and may account for some of the bizarre vehemence in discussions of this topic.
4) A clip watch is the best solution, anyone not using one is wrong -- I have spoken. :-)