When people who seem to be closed-minded, (or contrary-minded) are encountered, a common reaction is to challenge that person's position, opinion, or belief. If the challenge is unsuccessful, why be frustrated with that person and their different opinion?
Why not just move on, rather than moving it up to a dispute, with all the associated discourse?

People are free to disagree, and have different opinions, and it most often has little impact on another person.

Why is there an overwhelming need to convince anyone of anything? Could it be the need to be right? Is real source of the frustration with your ego, not with the other person, or what you see as an incorrect belief?

Why not just reply, "You could be right", and move on? After all, from the other person's perspective, they are also right. What does it matter to you if they hold different opinions or beliefs, on what is right?

A good question to ask is, "What is likelihood that anything I say will have any impact on the other persons opinion?" If the answer is none,or extremely low, why bother?