In the majority of commercial, mass consumption projects I've worked on, success or failure of "the groove" depends on the artists' ability to play to a click.
It sounds a bit backwards, but often it's that metronome-solid tempo that allows us to bob our heads and move to the beat, and changes in tempo (however subtle), may throw off your audience's ability to just lock in and enjoy. Grab any pop/rock/country/hip hop/edm track from the last 15 years and I'd bet you that the majority are an exact BPM from start to finish.
I always start out writing to a click/grid, and make sure drums lock to that. Then if all the other players use the drums as their "click," their natural tempo variations throughout the track will allow things to sound tight, but not robotic.
Of course this will change dramatically when working with jazz/classical, but the funny thing is that if you ASKED those players to lock to a click, they could do it. Every. Time.