Here is a paper which analyses the coupling of sliding and spinning motion with thin disks, and why both motions come to a stop at the same time. The paper establishes a mathematical model and reports results of experiments with a CD on a nylon surface which test the theory.
Edit in response to comments: The mathematical model assumes a flat disk with uniform mass distribution and explicitly ignores effects from its finite height (a non-uniform normal force leading to curved trajectories). The experiments use a CD as an approximation. A top may behave differently, although I can observe that the tip of a top "wanders" along the surface as a result of forces (like precession, a puff of air or an unevenness in the surface) which would by far not suffice to move it if it were not rotating.
The underlying reason for this interaction between spinning and sliding is that dynamic friction is independent of the speed of movement; it only depends on the normal force (here the gravitational force) and material constants. Its direction at every surface point opposes the direction of that point's velocity. With a fast-rotating, sliding object all velocity vectors are almost completely tangential because the rotational part is dominating the vectors. The magnitude of the vectors is very high, but irrelevant: The frictional force does not depend on it. Consequently, the frictional forces are almost entirely tangential as well; they mostly cancel each other out with respect to the forward motion and slow only the rotation.1
In effect, the frictional torque (which slows the rotation) is higher than the linear frictional force (which stops the forward motion) if the rotation is fast compared to the forward motion — and vice versa. This is why the "faster" one of the two movements is braked more, until they align and come to a stop together. The figure below (p. 2 of the paper) shows that interdependence. $\epsilon$ is the quotient of the forward motion and the angular motion, $v/R\omega$. For little linear motion but fast rotation the friction torque dominates (the left side of figure (a)), and for fast linear motion with little rotation the linear friction dominates (right side of the figure):
1 This is somewhat unusual: We often "dissect" velocities or forces into their constituents and consider them individually, independently. In this case though the lateral component influences the longitudinal friction because it changes the direction of the vector, and vice versa: Because the friction in a given direction does not depend on the magnitude of the vector component in that direction. The magnitude of that component is constant, the friction is not. Pretty counter-intuitive.