A push to listen button would be a small help but a lot of the handling noise is from the pipe being nudged if the patient moves (which they always seem to do). The button would need to be very light touch because pressing the diaphragm too firmly to the skin attenuates the lower frequencies. Seasoned clinicians deliberately modify the pressure depending on what they are trying to listen to (e.g. much lighter pressure if trying to auscultate valve sounds compared to breath sounds). The button would need to be waterproof and detergent wipe proof because I wipe it between patients.
(A "push to mute patient" button would be amazing, because another thing that patients love to do is talk when the stethoscope is on their chest!)