However, it doesn't end up any closer to the centre, since it already has a tangential velocity, which moves it from its original position; we therefore have to combine two infinitesimal displacement vectors, one tangential, and one radial. When we add those up, we find that the mass has moved "down" (i.e. closer to the centre) and "left" (assuming anti-clockwise rotation), and the resultant vector puts the mass back on the circumference of the circle, but a little more anti-clockwise.
In the absence of the radial motion, or if the radial speed is too small, the mass would fall towards the centre. You can imagine also a satellite in orbit around the earth. Its "fall" towards the earth each second (of about 5 m (
s=0.5gt2) is combined with its travel around the earth each second (of about 8000 m), to keep it the same distance from the centre of the earth, and the same height above the surface (which is curving away underneath it, of course).
If the satellite starts to lose tangential speed, then the combined displacement vector moves it closer to the earth each second, and eventually it hits the earth.