Having the other party not hear me isn't enough. Old-fashioned phones would mute both parties.
If I want to listen to something while I have a person on hold, I would not want to hear them.
I understand this is important for you, DSfromNY. We want to make sure we are on the same page to find a solution. When you tap on the mute button, the other caller(s) can still hear you? What is the make and model of your device?