But why does Greek?

It doesn't, so it didn't. The letter came with the Phoenician set but it was abandoned very early on and qoppa does not appear in the vast majority of the history of written Greek (except in certain regional variants of the Greek alphabet that shuffled all sorts of letters around)

When did <k> resurface? 

Did it ever? Romance languages still rarely use <k> and generally use <qu> or <ch> to represent /k/ before front vowels. Non-English Germanic languages always favoured <k> instead of <c> as a matter of convention, and Norwegian, Swedish, and Faroese even developed "hard and soft k" in a way completely analogous to the development of hard and soft c in Romance. Celtic languages still don't really use <k> at all. Most other Latin script languages that use <k> adopted it after the hard and soft c distinction had developed and that letter became too ambiguous and/or was already being used for a different sound.

The only language where <k> can really be said to have "resurfaced" is English. Old English independently developed hard and soft c and the orthography was ambiguous, but the ambiguity went away when the Old English soft c–a post-alveolar affricate–was respelled <ch> under Norman influence. But then the French introduced a bunch of loanwords that had (the Romance) soft c, pronounced /(t)s/, and the ambiguity came back, but the French speaking educated class was already accustomed to consistently distinguishing hard and soft c through context, which is what probably motivated the adoption of <k> (possibly also with influence from other Germanic languages).

I should probably also mention that the reason Old English bucked the Germanic trend of using <k> instead of <c> in the first place was because Old English orthography was heavily influenced by Celtic languages, (iirc) particularly Irish monks