Why would the other players be able to follow a click, but not the drummer?
Yes, ideally it would be great if everyone had the same sense of following a tempo regulator. But that's often/usually not the case, and when a player hears two drummers - a click and a drummer - and has to decide which to follow, their inability to all make the same choice as the next guy is magnified. At least if everyone hears just one drummer - whether click or drummer - there are no options. If they get off the beat, they only have one choice to get back on.
I've been recording or performing to clicks for 30 years, and have never had another player want to hear the click if I'm also playing.
Bermuda
Im nowhere near as good as you, which is why your bandmates can trust you as though you were a click. I have just joined my band, and nearly all of it is out of 4 and usually its an alternating meter kind of deal....I;m still learning all of it, and I need the metronome crutch way more than the guys who wrote this stuff.
Sometimes with all the irregular metering stuff I'll start a bar a beat early or a beat late and it really throws my timing off trying to catch up to where I should be, it really helps to have that click for the whole band so we can count the bar through all the syncopations without trainwrecking over a couple slip ups on my end (talking about rehearsal, not ready for shows just yet)
From my perspective, I know there are some parts of our setlist where the tempo picks up just because of the momentum of the piece. One part of one of our songs theres an instrumental section where I stop thinking about things like I should and lose track of the basics. I really like the guitar parts that the guys are playing and I just immerse myself in it and try to make the audience feel the same thing I hear as I play it by channeling that involvement into it. Everytime we hit it, I jump the tempo up 10bpm or so when the double bass comes in. I notice about 4 counts into it the click and the rest of the music just breaks away from where I thought it was and I need to readjust, the other guys are right on top of the clicktrack which makes it way easier to get back on.
IF we do it without the click, it sounds seamless and perfect and all that magical stuff, because its a natural pull. BUT we want it to be 100% solid like the recording, every time we do it, and if they stay right on the click for that part while I rush a little bit, then I can notice the mistake and readjust. It would be a horrible headache trying to get back on the click in just my headphones and get the other guys to follow..it would feel really awkward. I've been getting pretty good with it, if I lose my focus and flam with the click I am back on beat by the next count its just really good to have everyone on the same page with the tempo.
Thats all just for rehearsal though, for performance everyone needs to have the click because I am not always playing, but i need to be back on my china cymbal in 12 counts, or 3 repetitions of a riff or whatever. I just think its best (for us learning drummers and local musicians who are trying to get where you are) to have as much metronome exposure as possible, solo practice, band practice, and performance. So then 30 years later, people don't need the click as long as one of us is playing to it. It can only help right?