I frequently see people say they think bands are great as everybody brings their own influences to make their own sound. This makes sense, but I think it can go too far as there needs to be some common ground.

I used to play with a drummer who had very similar music tastes as me and it was f*cking amazing. We played all our favorite songs and we even wrote songs that didn't sound like anything else. Now I moved countries and I had two guys come over, one who was into classic rock, and one who was in to pop (he's a singer), and I am into hard rock and punk. We tried to write our own song and it didn't work at all. Thinking back though, why would it have? These guys don't know hard rock or punk so they can't work with it. Sure everyone brought their own influences, but it sounded like crap.

I'm not sure why I felt the need to say all this, I just recently experienced how messed up this one thing I always heard was and wanted to say something. What do you guys think?
If you can't work together, then kicking them out is kind of the only option in my opinion.
I agree. I've been playing and writing heavy metal (like power metal and thrash metal) with my friend/bassist for the past 3 years. We're both into metal, but he also liked listening to classical while i've listened to punk and some death metal. However, we both enjoy enough similar music that it's fun to play together.
And i agree with you. if backgrounds are too different, it aint gonna work out.
You should see what the results of this are.

1) Is your music too diverse. For example: One songs generic pure thrash, while the other is nu metal (aka genres that mostly hate each other)
Or even more extreme to where one's metal/rock and the others rap (aka genres that mostly hate eachother).

2) Does the songs themselves sound like they're stitched together all weird because it's being written by however many completely different genres and not flow well.

3) If one or two band members mostly write the music (most common case in my experience) then does this issue ever really ever come up?
Ultimately, you have to be excited about the synergy of your different musical styles.

Are you excited about mixing of hard rock, classic rock, and pop? Is there a middle ground there that you love the idea of exploring?

If so, it can work. If not, none of you are likely to be happy.
I think the key is having common ground.
My main band has a variety of influences. I started learning classical guitar and African drums. Our singer/other guitarist took years of classical piano before teaching himself guitar, and knows a lot of jazz theory. Our drummer is into punk, pop, pop-punk, and rap, and draws his rhythms from there.
We have a common ground in heavy music though, and that's why it works. We all bring different flavors from our other interests into what we can create together.
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The problem is most likely inexperience.

Seriously, you're saying that:

- Classic Rock

- Pop

- Hard rock + punk

Cannot mix? That's just silly.

It's more likely that you guys were just starting out, and didn't know how to start writing. Nobody saying "here's a chord progression/riff, I think it'll be really cool for a verse" and everyone starting to get the structure down. Alternatively nobody says "I have a full song here, I'd like you guys to learn it".
And no, Guitar Hero will not help. Even on expert. Really.
I think it's because rather than blending it all together, you're playing them over each other. Which is why it sounds like crap.