I'm a musician first and foremost and I have spent decades learning groove and feel. Timing is such a fundamental element to playing music, shouldn't it be left to the musician rather than the mix engineer?
I don't accept the argument that playing to a click means a trade-off with feel, or groove, as I play with drummers that use clicks but have an abundance of the former. Similarly, there are plenty of drummers that don't play to click that have no feel, or groove, so one wonders that their argument is?
I play in one particular tribute where the drums/bass are interlocked, more so than most bands, and without the groove it simply wouldn't work: even the most acute pair of ears wouldn't notice that there's a click in place, that's all down to the drummer. Just because there's a click that doesn't mean he can't push/pull the rhythm, of course he can.