Many good opinions and points so far, keep them coming.
Click track as an instructional method to learn your craft, it really is indispensable. And it amazes me how a lot of musicians I have met, apparently have decided to skip using one for training purposes...
True, not every song benefits from having a click track, however, be it live or studio. Some tracks really benefit from the "free time" and elastic timing. This however, is a band thing and it has to be done together as a unit.
Now, many drummers confuse "groove" and "shady timing", and thus, are offended by or uneasy playing to a click track. And therefore, "click kills the feel". But, for the rest of us, drummer's "groove" is about trying to guess where his 1 will land, which doesn't make for tight interplay.
I did a tour last summer (mixing monitors) for an act that features a phenomenal professional drummer, the rest of the band pro's, too. About half of the songs were done with a click track, and it amazed me every night how much room there is for interpretation even with click. To acclaim that click makes you stiff and boring, simply is not true. The click is a guide that doesn't kill your groove, and, IMO, should be considered so.