For some reason the counter doesn't go forward anymore, but, the film is rolling, and the film is being exposed, but the LCD constantly shows the "C250" message, which indicates that it's firing the shutter at a fixed speed of 250, until the first two frames of film are properly rolled. The problem is that no matter how many times I expose and advance the film, counter stays behind 1, and LCD shows "C25"
One possibility is that the little metal tab that depresses when you close the film door may not be engaged properly by the door when closed (e.g. if the door is bent slightly). This 'finger' is a small metal tab located in the dust seal track just below and to the left of the film advance crank. If this tab is not depressed fully, the film counter will not advance when you wind on. To test this, take the (empty) camera and open the back, then use a small screwdriver or similar to hold the metal tab down while you shoot and advance a few frames. If you do this and the frame counter still does not work, then it's possible (as you suggest) that the film advance is damaged and not working properly. In this case, it requires professional attention to fix, though spare parts for this model are long discontinued. Even if you have some camera repair experience I do not suggest removing the top plate and attempting repairs on this camera yourself, as you will find delicate light meter wires and a complicated knob/viewfinder pulley system to contend with (one of these pulleys is possibly where the fault lies).
Listen for the mirror and the shutter, i.e. the stuff that makes the noise when you hit the shutter button.