"Tick, tock" is actually the exact mechanical sound my parents' grandfather clock makes, so I might use it if I was describing it. However, the "clink, clink, clink" of breaking class makes no sense to me. Why is it written 3 times? Glass makes one chaotic mass of noise as it shatters and then the deed is done. The nearest sound I could get to it is what they say in my other language, Turkish--"shangüür", but obviously that couldn't be used. I think onomatopoeia is something to use very sparingly and according to the tone and style of the writing...
A bit of onomatopoeia is fine (a lot of normal words in English are onomatopoeic, and we don't even notice it). But "clink" is the sound of glasses being used normally, not of something being smashed, so you need the right onomatopoeia. (And I'd check that "sit" is the right aspect of the right verb, but you didn't want general writing help, did you?)
Son #2 used to call a clock a 'ticky-tocky" and a pair of scissors: 'snippy snappies". Maybe not quite, but descriptive anyway. He grew out of it of course.... Hetty
But that's a bad use of onomatopoeia. Onomatopoeia in itself takes skill to use, so in itself I'd say a good writer WOULD use it. You say the impact should be carried in the writing alone - well, onomatopoeia is a writing device, so I'm not sure why you're treating them as separate things. It's like saying "Well, my poem might be stronger if it has no meter and didn't rhyme and perhaps no verses either. I feel a truly good poet should be able to write a good poem without these tools." Essentially, what you're saying doesn't make sense, to me anyway. Knowing how to use these devices, and when, is part and parcel of being a good writer. To give an example: "The snake slithered, seeing for the first time the sumptuous servings of rat." Now that's not entirely onomatopoeia - I don't personally know what on earth it is. I'm not saying that sentence there is a brilliant sentence, but that's just an example of how I'd personally use onomatopoeia mixed up with other poetic devices. I'm mixing the s-s-s sounds with the st in first and the s and t-s in sumptuous. Notice sumptuous and servings both begin and end with an S. That's how I'd play around with sounds and rhythm personally. Again, not saying it's a wonderful sentence, but it is more sophisticated than "clink, clink, clink". "Clink, clink, clink"? This is more of a primary or secondary school example which, in itself, is fine, but as writers aspiring to be published, unless you're talking about children's books, I'd personally explore a little more. The problem isn't onomatopoeia, but how onomatopoeia is used.
Alliteration? You can use non-onomatopoeic words for alliteration purposes as well. Like "the sun shone." You've got two sibilants and you could say there's alliteration, but no onomatopoeia, as far as I've understood the terms. Not that it matters that much, as long as the prose evokes whatever the writer wishes it to evoke, and, like @digitig said, you use the appropriate one.
Haha my bad, I got them mixed up! I still think that onomatopoeia could be used well though and probably depends a whole lot on the tone your writing already carries sometimes it could be used to make something sound creepy.
I'll give my honest opinion of the effect it had on me. I liked the little snippet. You set a scene using few words, and I found myself happily inside the scene. But lose the "clinks". They slow it down.
My biggest issue with this is the culture barrier. For instance, clink clink to me is not the sound of glass, even though I on a logical level know what it's supposed to be. That would be klirr klirr. Clink clink makes me thing more of klank klank, which in my culture is more of a metallic sound.
I think it all really comes down to "killing your darlings"s during editing. Onomatopoeia works in more subtle ways, and is best exemplified when it is refined during revision. An example of good, surprising onomatopoeia comes from Raymond Carver (I think; I've read a lot of stories this semester) referring to a car driving across wet pavement after it's finished raining. He says, "... a car rubbered by..." That small phrase just really hit home for me.
I use them in the narrative on the odd occasion, though I can go entire short stories without using them. They are by no means missed.