Never use that phrase with me again, please. It is a most insulting phrase.
It is saying “You are wrong, and you are not even worth engaging with to try and change your mind. I am always willing to listen to a strong argument. I do not agree to disagree with you; I agree to hold on to my opinions until shown a superior one. I agree to keep trying to change your mind until it is changed, or until you provide a better argument.
That phrase has been used on me too many times to dismiss my concerns. It is associated with one of the worst job interviews of my career.
Either argue your point or stay silent.
I do not agree to disagree with you in perpetuity. I disagree with you now. I am willing to put in the effort to bring you around to my view point. I am willing to listen to your point of view.
If you are not willing to engage with me, do not engage with me.
I am far more likely to say “I am not willing to pursue this argument.” Which I often will do for things that may be too painful to discuss. That is usually politics, religion or other sensitive matters like that. But that is almost always an issue of self-preservation.
This phrase has not value and you should excise it from your lexicon.
If your disagree, you better have your argument in order.