You can't avoid the time. It's in your car, on every bit of electronic kit at home, on the side of buildings, on all sorts of displays, everywhere. Why anyone NEEDS a watch beats me.
There speaks a man who lives in a city. And has far too much electronic clutter around. The only thing which would tell me the time where I am apart from my watch (and the computer I'm typing on, which is often tucked away in sleep mode or off) is a big old-fashioned clock on the wall. In most rooms in my house it would be only my watch. Playing with my son outside the house or taking him down the park, the only alternative to a watch might be a sextant. Walking around town at lunchtime I'm struggling to think where I'd be able to see what time it was without going to a lot of trouble. If I'm running or biking up on the hills, the only available clock is somewhere I tend not to go on a lunchtime trip.
Like others I don't carry a mobile phone all the time – in fact it's pretty rare that I do. I suppose if you put it like that I don't NEED a watch – I could carry some other gadget with me solely for a function it wasn't designed for and which it's not as good at as a watch. Alternatively I could be late for post-lunchtime meetings.
Watches are useful, but perhaps more of a fashion accessory/jewellery these days, considering how we have the time on our 'phones, computers, TV recorders, cookers, in cars, etc.
I don't tend to carry a cooker, TV recorder or car around with me.
I've gone for a while without wearing a watch when I've had wrist injuries, but it always goes back on afterwards. FWIW my day-to-day watch is a £300 Suunto HRM!