I was going to say - didn't some sort of popular vote play a role in the whole ELCA/gay marriage issue (or was it just gay ordination?).
The Catholic belief is that the pope is infallible in matters of faith and morals, and even these seem to have some qualifiers (e.g. gravity of the statement, repetition of a statement, etc.). So it's not as if there's a belief that the pope can do no wrong. Obviously the popes are sinners like the rest of us, but that doesn't take away from the office he holds, according to Catholic belief.
Who can "throw the pope out"? Funny you ask that question. That is a kind of issue we have discussed in my class I'm taking. I don't recall the exact the parameters of the discussion, but the issue of conciliarism emerged in the 15th century and so there is some merit to the bishops exerting power in this regard. However, I am no theologian and don't want you to quote me on this; just know that at least in the past, the issue has come up; it could be revisited in the future, but that would entail a lot of bad things happening first. The Council of Constance is where this issue played out.
We've also discussed the emergence of the cardinals. They seem to have been chosen from the titular churches of Rome (some of the earliest ones) and some nearby regions. They helped to operate the Roman Curia (court) and had a role in jurisdictional matters, which you may know got to be quite extensive in the Middle Ages because of the expanse of the Church. They also had the ability to elect new popes, which seems like an objectively good thing so as to regularize the system and prevent state interference.
I guess the (objective) problem that I see with more democratic ways has to do with the checks on that kind of system. For example, with the issue of birth control, the pope (Paul VI) in the 1960s condemned this as morally wrong by way of encyclical, even though there was opposition in the Church by "
cardinals, bishops, priests...". Now, there is only one right answer - it can't be both morally permissible and morally wrong at the same time. With "democratic" church governance, what would have happened? Perhaps a split somewhere, with one side in the right and one side in the wrong. What would have happened to Christ's Truth? Now, Catholics can say definitively that other Catholics who still espouse the use of birth control are speaking in error even if they sincerely believe that they are right.