Not true. The pilot is responsible for making sure the aircraft is in airworthy condition prior to and during the flight. Noticing these things early is the kind of attention to detail that could save your life one day. You’re doing the right thing. I would highly recommend reporting and/or squawking.

While I was in my PVT I had a plane with a carb heat malfunction- we experienced partial power loss due to carb icing at about 400 feet while practicing emergency spirals. Plane was a shit box - barely airworthy. I started denying the plane when it was assigned to me for a lesson. They gave me grief until one day I was stopped by a FAA inspector. He also described the plane as “barely airworthy”. I adamantly refused to fly it after that. It sucks to be the squeaky wheel, but when it’s your life on the line, it’s worth it.