Wouldn’t the younger generation at least have seen analog clocks, even if they’ve never learned how to read them? So they might at least have noticed the direction of movement of the second hand.
When have digital watches, or mobile phones, ever been cool, or fashionable for more than five minutes? ISTM that if one is going to bother to get a clock or watch (not a given), it is probably not going to be digital.
I’m not sure how true that is. I’ve taught kids to read clocks, and I’ve seen at least three curricula for doing so, and in none of these was there any conversation about angles: it was all about looking at where the tips of the clock hands pointed. I’ve also taught kids to evaluate angles, and in no curriculum I’ve ever see was there more than a passing reference to clock hands. I just don’t know that these concepts are connected in the way that you’re suggesting.
Nor am I convinced it’d be helpful. A clock’s hands form a right angle at 3:00, sure, and also at 9:00. How does recognizing that help you tell time? They also form a right angle at various other time–somewhere around 4:06, and 5:12, and so on–and trying to introduce that fact would be super confusing to little brains who are still trying to figure out why you multiply the minute hand by 5 but not the hour hand, and why it’s the big numbers times five but not the little notches between the big numbers, and who cannot for the love of Christ remember that when the hour hand is almost at a big number, the hour is the previous number.
I don’t want to get too far afield–spreadsheets are just one example of something that might be preferable to analog clocks in a modern math curriculum–but I think spreadsheets should be taught in much more depth than simply reading tabulated data. Formulae, sorting/filtering data, and creating charts are all phenomenally useful and not terribly difficult to learn. But I’ve talked about it with my daughter, a high school junior who’s taking advanced math courses, and she’s never been taught the first thing about spreadsheets.
Back in school, I studied French and Russian. When asked why, my answer was that it was better to be able to speak with more people than those who only spoke English.
It’s the same with analogue clocks. If you can read them, then you’ll be able to read more clocks than you can if you can only read digital.
Besides, why wouldn’t anybody not want to know what time it was by looking at Big Ben in London (yes, I know that Big Ben is the name of a bell in the tower, but just go with it, okay?), or the analogue clock in Grand Central Station? Both are famous, and both keep accurate time. Why not learn those, as well as digital? More languages, more communication, more understanding, as I implied.
Why not learn ancient Greek? How to poach an egg? Cuneiform? Slide rules? Guitar amplifier repair? The history of coffee?
All of these are worthy things to learn. But we have limited time in K12 education, and the question, again, isn’t whether a topic is worthy; it’s whether it’s the best use of that limited time, or whether there’s something better that kids could be learning.
Umm, yeah. There are still analog clocks and watches everywhere I look.
“Far out in the uncharted backwaters of the unfashionable end of the western spiral arm of the Galaxy lies a small unregarded yellow sun. Orbiting this at a distance of roughly ninety-two million miles is an utterly insignificant little blue green planet whose ape-descended life forms are so amazingly primitive that they still think digital watches are a pretty neat idea.” -Douglas Adams
They spent half a day teaching us how to tell time when I was in 2nd grade. 4 hours teaching a useful, lifetime skill.
What are they teaching second graders today in that amount of time that is a more useful skill?
The OP’s assertion that there aren’t any analog clocks anywhere, or that it’s not a useful skill to have is absolute horseshit.
I am skeptical that you are remembering this accurately.
Thank you. I screwed that up hard. 
I had forgotten which way is already used in math. I should have looked it up or added a disclaimer like: “Or maybe the opposite if there’s already an established standard somewhere.” I almost added that disclaimer, but skipped it. Oops.
So yeah, positive is anti-clockwise & negative is clockwise as you rightly say. We have enough problems with electricity “positive” and “negative” and with magnetic “north” and “south”. There’s no reason to introduce a new instance of sign backwardness when we already know better.
Which also suggests that we should have built the original clocks to rotate anti-clockwise … err … in the positive direction. 
Which is kind of funny. It just seems so natural to have the numbers increasing in a clockwise way, but that’s purely convention. Once we built clocks that rotated clockwise, the direction of increasing became fixed for all sorts of arc-like / angular measurements. The tail wags the dog. It was an arbitrary decision then that carries down through the ages in far greater consequence than that first clock-maker could ever know.
The clockwise direction of a clock’s hands is arguably not arbitrary, but follows the direction of the original natural clock, the sundial (in the northern hemisphere, where clocks were invented). As the sun moves from east to west, the shadow on the dial moves from the right, to forward, then to the left.
We already live in that universe. 
The word “clock” refers to the bell, which is the original user interface of time keeping devices. If you want to know the time, count the number of times the clock (that is, the “bell” in today’s parlance) rang. One ring for one o’clock, two for two, etc.
Eventually, many clocks got a face with a hand, and then sometimes more than one. And the word “clock” started to refer to the timekeeping device, or even just the clock face. And we’ve mostly forgotten that a clock is a bell to tell us the time.
There are clocks that actually do this, usually sold as novelty items:
This one was even built with reversed numbers, so it will look correctly when seen reflected in a mirror:
ETA: get one (or both) if you REALLY want to exercise your brain! 
Keep in mind that when Adams wrote that, digital watches used LEDs and only displayed the time when you pressed a button.
As I mentioned in the other analog clock thread, I have never worn or wanted an analog watch. They have absolutely no appeal to me whatsoever.
Why do you want to know the time? Do you want to be able to recite the time to one minute precision? Or do you want to know how much time is left until your 2:30 meeting?
For me, the later is more often my goal, and i find an analog clock easier, i just look at the distance, i don’t have to do subtraction.
That being said,
- i don’t think reading a clock face is a critical skill that every citizen needs.
- i did make my children learn to read clock faces.
The college students i interact with can read analog clocks. But many of them haven’t learned to write by hand. I mean, they can carefully form letters (and read Aunt Mary’s note) but they haven’t practiced the skill of writing letters enough to become fluent at it. Some of them visibly struggle to write their name and email address for me.
But clock-reading was a kindergarten skill when i went to school, and i don’t think it would be appropriate to introduce spreadsheets until at least 4th grade, and probably not until 6th. So they don’t compete for that limited time.
You ask a good question, though.
A younger friend suggested that “clockwise” and “counter clockwise” were selected arbitrarily. I pointed out that it drives from sundials, and was not not arbitrary. He found that interesting, but that’s certainly not something kids need to learn.
Lol, mine does. Because I find it easier to use that way.
Whenever I have shopped for a digital watch I always choose one that includes SECONDS on the main display. Forget the ones that show only the hour and minute.
The analog watch on the home screen of my phone includes a second hand.
I was addressing your question if someone wants to know the time down to the minute. No, for me down to the minute isn’t good enough.
I tried to poach an egg once. The gamete warden caught me.
I, also, sometimes want to know the second. That’s one of the benefits of an analog clock, it’s easy to toss in a second hand without making it bigger.