Fortunately, depending on the style of mediation used, that's not the way it works. I've seen it happen. I think the mediator's peculiar role as a neutral is the key to what makes the process work. Because the mediator is neutral, he or she can throw out ideas that both sides may have thought about but neither side wants to suggest. The mediator also develops credibility with both parties that is usually not there for either party or for either party's lawyer.
Lawyers love to make proposals and defend positions.
A good mediator will steer the parties away from proposals and positions and toward accomplishing goals. There have been many times when I've been in negotiations representing one of the parties, and I've longed for a mediator. I knew that a suggestion needed to be made, that the other party needed to make it, and that they hadn't figured out that they needed to make it. And I knew that it might connote weakness for me to make it. A good divorce mediator would have stepped in and gotten the conversation going.