Butter I think the reason Claire/Clare/etc didn't weather the storm so well is because it's a relatively new name, unlike Sarah and Emma. The newer names tend to either fade or join the ranks of other cyclical names. It's basically a good example of exactly what the OP is trying to avoid - a faddy name.
A friend of my Mums, born in the late 1930s is a Clare, and I'd say she must have been part of the very early uptake of the name! Given that it really peaked in the 1970s.
I reckon Claire will have its day in the sun again, though. If you look at it without any preconceptions - as people will do in a couple of generations' time - it's a lovely name. Simple, elegant, unfrilly but feminine.
The fashion at the movement is for very feminine names, IMO. When we move away from the cutesy, twee 'ie/y' ending names and the 'a' ending names, Claire will have its place again. Other more solid names (names ending in consonants/consonant sounds!) will come back as well - Joanne, Karen, Helen, etc - all these names are eschewed right now, as the fashion is for much more frilly names. Obviously not everyone succumbs to fashion, but this is the trend at the moment, if you ask me.
I also agree that you can't say Sarah isn't a timeless name - I guess it just went through such a resurgence at one period, that it does maybe sound a bit 'dated' for want of a better word, for one particular generation (i.e. Gen X, born in the 70s). For other people it hasn't been tarnished through sheer overuse.