I'm in my late 20s, and I don't have a voicemail box. I hate vm. People always take way too long to get to the point, and usually I can't even tell what they're saying anyway. They'll quickly mumble a phone number for me to call, so I have to listen to their three-minute message multiple times. Listening to voicemail requires sitting on the phone with a robot for about ten minutes. I can not remember the keys to operate voicemail systems, with the exception of 7 for delete.
I'm fairly sure that voicemail is a ploy by the telephone companies to ensure that they can bill for otherwise unanswered calls. It wouldn't be worth the maintenance cost otherwise. On my T-Mobile line, I can't turn it off, or even set the forwarding delay to more than 20 seconds.
Instead, I set my voicemail forwarding number to my home phone. That gets me a few things I like. Incoming calls when I'm home, I can forward to the landline, which has no compression. So I can actually understand my friends. And, if someone calls me at night, it's going to wake me up eventually. This is important to me because when someone calls rather than texting, it's usually both important and urgent.
It seems likely to me that your son has intentionally allowed his voicemail box to fill up, in order to not get irritating "You have a new voice mail message!" notifications every few days.
Duncan