Unless these are co-workers who have to put up with each other's methods of communication, I honestly don't see how these are "a group of 4 people who communicate fairly regularly". And if they are co-workers, well at work, you suck it up and you communicate as your employer instructs you to, even if its a method you hate.
My best friend and I are so close because we communicate similarly. My DH and I worked as a couple because we communicate similarly. The family members I have who communicate similarly to me are the ones I'm closest to these days, even if 15-20 years ago things might have been different.
I have friends and family who are phone people, and i don't shun them or hate them or anything. But the reality is I am not. I don't use VM and I don't initial calls (or leave VMs) unless I have no other choice. So I end up not being as close to the phone folks. And its not just because of me, on my end. They all have computers and as far as I know they all have texting. They are also choosing to not text, use email, or use Facebook messenger, just as much as I'm choosing to not use phoning & VM. And the reality is the relationships begin to fade a bit or they never really form well to begin with.
And the reality is I do use plenty of forms of communication regularly. I check my email multiple times a day, from my computer or from my cell, I also check my Facebook at least once if not 4 times a day, from my computer or via my cell - and I can check from any internet connected computer really. I check my cell daily, usually a few times. And I will answer my phone if i hear it ringing and I'm available to speak. So its not like its hard to get a hold of me, to get me a message or to get an answer from me, or have a chat with me despite me not using VM. So if someone insisted on using VM, despite my outgoing message asking to please not leave one, and despite the fact that it usually takes me days to listen to the VM if I do at all (I would probably just call back "I saw I missed your call, whats up?"), then yeah, messages will get lost and the friendship won't thrive. In some ways its sad, but in others it just practical.
I think the friendships should
fade. It makes sense that you'd form closer relationships
to those that can conform to your style of communication. But, I don't think it's rude not to. Your disdain for VM, and refusal to use it, is absolutely your prerogative. If you don't think it's rude for you to tell these potential friends that they either hang up the phone and text you when you don't answer or lose you as a friend, I don't think it's rude for them to say "That's a complete inconvenience for me. I don't want to have to push several more buttons and take up a several more seconds to leave you a message, simply to save you the trouble of spending a few seconds typing in a password. This friendship isn't worth it." (And if listening to a ten second message is so over the top painful, that doesn't say much for the conversation you would have had, had you answered the phone).
For me, that type of friendship, where one is so picky about which of the many options of leaving a message is used, is too high maintenance for me to even attempt to maintain. I think it's more rude to insist that your "friends" conform to your preferences and yet not be willing to budge an inch and listen to a VM when they already went through the trouble of trying to call you.
Yet, I do understand that relationships
with people who communicate the way you do are formed naturally. I'm just not going to say that it's rude to leave a VM just because you don't like it.
Again, I find this thread very similar to the reading a text vs. reading an email thread. And I think, for me, the conclusion of this one is the same as that. It's not rude for people to choose a form of communication that is common and readily available to try to communicate with you. If you absolutely cannot bring yourself to communicate that way, then don't. Yes, those people will likely feel that it isn't worth the extra effort to get a message across to you and will stop trying. Those relationships
will fail. It's not their fault and they are not rude because that happened. It's a consequence of your own behavior pattern that caused it.