A complementary position is that just because something is 'grossly insensitive' doesn't mean it isn't both a kindness and exactly the right thing to do. Humans learn from unpleasant things. Especially targeted unpleasant things. So 'got to be cruel to be kind' often applies.

If your true and actual reason for breaking up with someone is that her breasts are too small, consider that (a) saying "It was because you were too clingy" may cause them to try and mess with an aspect of their personality that doesn't even need fixing, and (b) total silence, which you may fondly imagine to be mercy, may result in her frantically imagining dozens of possible flaws all of which she tries desperately to correct, just on the off-chance it was that one. As opposed to, say, looking for a guy who's into smaller breasts next time.

Tangent: The tricky thing is that often "because you were too clingy" will technically be the real reason, just not the most useful part of the causal chain to select. If she had bigger breasts that will change both how 'clingy' any given behavior seems and how much attraction to her you exhibit which in turn influences how clingy she is likely to be. So sometimes even 'real' reasons can be a cop-out!

Maybe I'm just being inordinately naive, but telling someone honestly, softly, and believably, your true reason for rejecting them, seems like it really should have certain advantages for them, if not for you.

That certainly seems likely for most cases.

Even bigger tangent: I can't think of many better ways to be broken up with than this! Seriously. It's (counter-intuitively) one of the least personally insulting break ups I've seen. Because pussy-footying around being 'sensitive' is in its own way just another kind of insult.