I probably have taught more kids to read “clock time,” both analog and digital, than just about any of the rest of you 
[I taught primary school for many years, where clock reading has traditionally been a significant part of the math curriculum, and now work part-time as an elementary math specialist.]
So, a few thoughts from someone what’s been in the trenches.
–My school has an analog clock in basically every classroom. Other schools I’ve been in (supervising student teachers, giving workshops, etc.) also have analog clocks in basically every classroom. These are just about the only analog clocks kids will see in public spaces: the time-and-temperature signs along the roads are all digital, the clock at our local ballpark is digital, etc. Of course time displays in cars and on phones tend to be digital, and it seems that more and more people who wear watches these days are wearing digital watches. I know I don’t have a single analog clock anywhere in my house, and I think the same is true for a lot of the kids I teach. --There is something kind of strange about teaching analog clocks in school, when school is just about the only place kids will ever experience them.
–Analog clocks do take some getting used to. One issue is that there’s a developmental thing going on, where kids have to look at the numbers when they are referencing the hour and the numbers-times-five when they are referencing the minute, and that is difficult. In my experience a lot of kids can’t really make sense of this (and read the hands consistently) till the middle or end of second grade. Even then it can take a fair amount of practice to get them used to the process.
–Digital clock reading is easy, but does lack context. I have never seen a child think of 5:57 (since this was an example given) as 5:00 in “digital world”; the real issue, I think, is that it isn’t at all obvious that 5:57 is actually “almost 6:00.” That’s the big challenge of teaching kids to read/understand digital clocks. Since the “magic number” [the base, essentially] is 60, and kids are used to 100, seeing 5:57 does not automatically ping their round-up meter the way seeing the number 597 would (oh, it’s almost 600).
–Related to this is that as long as we continue to use expressions like “quarter after five” and “ten to three,” digital clocks don’t help kids understand the meaning. [I keep expecting these locutions to go out of style. So far they don’t seem to be. i’M not sure why.]
–Getting back to a time like 5:55, kids will certainly see that in analog clock world this is “almost 6 o’clock.” The trouble comes when they are asked to write the time shown on the clock: huge numbers of 2nd graders will write “6:55.” It’s difficult to see that the hand is just before the six, and hard to remember that this means the time is not quite six. It can take a lot of, um, time to help kids get this straight.
–Analog clock reading does indeed have a lot of mathematics in it. You have to be able to count by fives, count by fives and ones, think in terms of fractions, etc. As long as we continue to teach analog clocks, that’s a side benefit. Of course there are many other ways to teach these concepts, which may be a better use of limited resources.
–Since someone mentioned teaching fractions through analog time; the series we are using right now is very big on that, but IMHO it confuses more kids than it helps. (One sample idea is that you can model 1/3 by shading 12 to 4, then model 1/4 by shading 4 to 7, then see that 1/3 + 1/4 = 7/12. The idea seems reasonable but as I say it doesn’t seem very effective in practice.)
For me, the bottom line is that teaching analog clock skills takes a big chunk of time and energy relative to its value. I think that time and energy can be spent more effectively on other topics. If I were in charge of the standards (I am not, alas), and of schools, I would replace all school analog clocks with digital versions, and drop the teaching of analog time. I’d include it only in the way that Roman numerals are sometimes included–as a kind of curiosity, useful for illustrating some specific ideas and for showing how things used to be, but not worth much beyond that.
You can probably guess how I feel about cursive… 