I think it's the way of the world now, you can't get away from it.
I don't think you'll find many top names who don't use either backing tracks, samples, pre-recorded parts or click tracks to some degree.
I was the keyboard player in a band for many years both original and covers. We never actually used any kind of backing track but the reason for this was:
The other band members didn't have a clue how to go about it and me being the keyboard player, well most of what I was playing could easily come from a backing track so I wasn't going to push the idea and do myself out of a job.
Most musicians and bands that we came across on the circuit though were using them, either openly or covertly.
It's amazing how many people in the audience either don't notice or couldn't care less that most of what they're hearing is pre-recorded and they might as well be listening to a cd sometimes.
I remember we were playing with a Wham tribute band one night and I was in the audience with my girlfriend and I said 'where's the sax player?' - what sax player she says - 'this is a sax solo, where's the sax player' is said - 'oh yeah' and she carries on dancing. I even caught Mr Gary Barlow miming his piano at the queens birthday bash when I was watching it. My wife who's a big Barlow fan was not impressed when I pointed it out!
Also supported a Bon Jovi tribute. They had one guitarist a bass palyer and a singer (no drummer) but close your eyes and you'd swear you were listening to the real Bon Jovi. I wonder how they did that?
In terms of using a drum machine or click track. practice, practice, practice is the key.
Make sure you can hear the damn thing, maybe use earphones/in ear monitors - but however you do it, make sure you can hear it.
The less complicated the drum track the better. If it's just a repeated rhythm for example, if the worst comes to the worst you can wait a bar and get back in time. If there's lots of changes programmed in you got to listen carefully and really stay on top of it.
Saying that, it's done all the time, just takes a bit of getting used to. You'll be amazed at how many drummers play out of time and change tempo when you start to play with a track. Even how much you yourself don't play rigidly, it can be quite strange at first.
Don't give it a second thought.
Good luck
Yodasdad