In this context, the phrase "tread water" means to perform an activity without making any progress. Literally, to tread water means to swim in place.

How can someone sit there and tread water at the same time? Not logical.

Tread water is to stay floating by moving your arms and legs while swimming without moving your location. I do not understand why the speaker has something they really want to say... But I think the person is comparing the activities of struggling to swim, being burned, and blown by a heavy wind to keeping her/his mouth shut.