It's not the priest doing the binding, though. He's saying that someone who gets up in front of a priest and professes marriage promises to another person is bound to that person until death. That is Catholic dogma.
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Besides, if a person comes in claiming the priest has to give him the sacraments, despite the priest believing that person is an adulterer, then it's that person that's trying to bind the priest's conscience, not the other way around.
That doesn't make any sense. Of course marriage is indissoluble and that's dogma. That's not the issue. What's at issue is whether or not a particular relationship was in fact a marriage in the first place. It has to do with the application of dogma to a particular situation.
Let's assume it's pre Vatican II. Couple has an annulment. But the priest decides it's no good (issued in error) and refuses the Sacraments to the couple. According to your line of reasoning, that would be permissible. Applying your first argument, he's not binding anything. And applying your next point, it's the couple trying to bind the priest's conscience, and not the other way around, by approaching him for the Sacraments. Would this be appropriate? Of course not. According to Canon Law, the faithful have a right to receive the Sacraments, so this priest would be unjustly withholding the Sacraments from them based on his private judgment.
And the same applies to post Vatican II annulments. Trad priests have no jurisdiction whatsoever, much less do they have the authority to make any determination regarding the validity or nullity of a given marriage. They can form an opinion, but they cannot impose it on the consciences of others who have a different opinion by punishing them for it by withholding the Sacraments.
I'll give you another example. I believe that NFP is a grave evil and that Pius XII was mistaken in his speech to the midwives when he opined that it was licit. If a couple came to me who were practicing NFP, I would advise them of my opinion that they were committing grave sin and jeopardizing their souls, and would make my case for for that opinion. But if they were not persuaded based on their argument that "Pius XII said it was permissible." (prescinding now regarding the vague notion of what constitute "serious" enough reason to use it), I would respond finally that, just make sure (examine your consciences) that you really believe that it's OK vs. rationalizing it to yourself by appealing to Pius XII, but at the end of the day I would not refuse them the Sacraments because they decided to use NFP. That would be for me to impose my own conscience and my own conclusions on them. Without the judgment of the Church behind such opinions, the priests overstep their bounds and unjustly impose their consciences on the faithful.