“Fair point” is two words that make everyone take the rest of your points more seriously.

Conceding something small signals that you are listening rather than waiting. It tells the room your agreement actually means something, and that is what gives your disagreement weight later on. People who never yield an inch become easy to discount, because their position clearly has little to do with the argument in front of them.

In the next conversation that tightens, find the one thing the other person got right and say so before you continue. Keep it short and genuine. Watch how much easier the rest of it becomes.