Yes, I do think you miss out of a lot of the Camino experience if you don’t stay in albergues at least some of the time. I’ve always stayed in a mix of albergue dorms and private rooms (in albergues or other accommodation), and while I enjoy the comforts of a private room very much, I would never miss out on what albergues offer. I don’t know that people can really say what they are missing if they never stayed in albergues or only a couple of times.
The communal dinners especially bring people together like nothing else. When you sit down to share a meal with someone and often have a glass of wine and good conversation, it’s a much deeper connection than just chatting on the trail, in a cafe, or even in the public areas of an albergue. Then when you see each other again, maybe days later, it’s like running into a long lost friend.
I have never seen much conversation go on in the actual dorm rooms. Often someone is resting or people just respect other’s privacy. And, btw, dorm rooms come in many sizes. You won’t necessarily be in a room with ten or more other people. It seems especially post Covid last year, there are many rooms in private albergues that only have three or four sets of bunks. Sometimes there is a small surcharge to stay in a smaller room. Occasionally there are only two sets of bunks, and in at least a few albergues along the CF, two beds per room or alcove. Some have capsules with privacy curtains. So don’t assume that you’ll be in a huge dorm with 200 others. In my three times on the CF, I have never experienced that, actually.
I do like taking a break from albergues, though, and I especially like having a private bathroom every now and then.
I also would point out that while your concern about maybe not finding a bed in an albergue is reasonable, the reverse could also happen. You could try to book a private room and find that they are all full, but the walk-in albergue that doesn’t take reservations has beds. This was the case for us last year many, many times. I highly recommend being as open-minded and flexible as possible about where you stay.