'Collect' is actually the original meaning of the Greek root λ-γ, whence λόγος, λέγω, etc. The later and more common senses came to be counting money, making an account of something, and then generally speaking and saying. (There are about a dozen important definitions of λόγος, with several dozen more minor senses. A fascinating, fascinating, word, made even more interesting by the Gospel of John's appropriation of it as the best description for Jesus Christ.)
In most of Ancient Greek, yes, if you wanted to say 'collect' you would use συλλέγω, although it could also be used for "I discuss". However, I don't think there's really any precedent for using συλλέγω in an English word for collection. -phile seems to be fairly well entrenched. I despise constructions such as 'keyophile', however; probably because I find the blatant mixing of roots, especially with random 'o' stuck in middle, somewhat ugly. I don't have a problem with mixing roots, I just like it to be a little more subtle. Key collector is fine.
Interestingly, the English word colLeCt is the only one where survives the Indo-European root which gave rise to the λ-γ construction in Greek.
Not a word. There is no usage at all for it on the Web (nor for *cagophilia), and it has no reasonable etymology. (Probably invented by someone arbitrarily deforming Old English cǽg.) Why not just say keyophile and be done with it? Or better, key collector?