I will go against the grain on this one and say that I think it is a good rule. It is a lot of effort to be the captain of a club if you do it properly and it should come with a few perks, one of which being able to play when you arrive at the tee rather than wait.
Of course, that does not necessarily mean that the captain should immediately take advantage of this on every occasion and you would think that jumping to the front of a 40 minute queue would be one of those times when discretion would be the better part of valour.
Sits fine with me. There are reasons (in my mind) that this unwritten rule should be there....
The captain may well be entertaining people from other clubs, sponsors, potential sponsors, new members, potential new members etc etc.
A quick nod to them and the question "do you want to go?" I've yet to see a capt abuse this and just jump in with their regular 4ball.
I don't have a problem of an unwritten rule that the Capt has priority on the first tee. He gives hours of his time every week, surely its not the end of the world to give him 10 mins of yours.
A decent Capt would only accept an invitation to play through if he has guests. He wants the club to be seen in the best light, and I dare say most club members would want the same.
A question; did you or your partners not know of the club "rule?"