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I'm not so sure. I get why it is the way it is. I think these numbers are called Highly composite numbers. That's why we got 12 and 60. (But 24 would be another one.)
But it comes with issues. As I said you start with the 12 and then the one. That's probably because the number zero had a complicated past. And now you have the clock going around twice a day and you need to prefix everything with am/pm. Or it's clear from the context.
I think the number Pi is the same complicated concept. Why half around the circle and you need to memorize all the '2's in the formulas? Why not make it once around the circle and use tau = 6.28... ?
So I think I can understand why we got there. But we have the number 0 nowadays. And electric light so we can stay up till 1am. So it seems an outdated concept to me to keep the 12 around. And if it were elegant, you wouldn't need to specify which turn of the clock you're talking about.
Once the kids of today finally can't read analog clocks any more, the 'cluttered' argument is a thing of the past, anyways.
(Edit: I think you can already see this. Ask someone young about the time. And an old person. Maybe this is why OP asked the question anyways. Someone below a certain age will probably read you back the exact numbers on their digital clock. I've never seen a person in their 60s do this. They always say 'quarter to nine' or 'a bit past six'. At least where I live (Germany) they do. And with the 24hour clock, we have both. You'd say 'come over for tea at 5 o'clock' but most people would write it down as 17:00, at least if it's somewhat formal.)