… Forget about thinking about anything spinning and any forces associated with rotating objects.  The point would be the same even if you ran a straight groove under the stylus of a cartridge in a tonearm with offset headshell.

The force of friction is along the line of motion.  Since the record groove is nearly perfectly tangent to the cantilever at the stylus position, the force of friction is pulling the stylus directly forward in line with the cantilever.

Now, get in front of the cartridge and look directly at the end of the cantilever, with your eye looking along the line of the cantilever.  Note that your line of sight does not pass through the pivot of the arm, it misses the pivot by a substantial distance to your right side.

To more graphically illustrate this, tape a light thread to the centerline of the headshell.  Cue the arm up.  Lightly pull the string and align it so it runs along the centerline of the headshell (which is also the centerline of the cantilever.

See how the imaginary extension of the string passes far to the front/right of the tonearm pivot?

That means that the force of friction can be resolved, through Vector analysis, (search "resolving vectors into components for the math) to have a component along the line running to the tonearm pivot (which causes no motion or side force on stylus) and along a line inward toward the spindle, perpendicular to the line running to the pivot.

So, if the stylus were not locked into the groove the arm would move inward.  To demonstrate this pull on the thread, while aligning it along the centerline of the headshell (again, this is the direction the stylus is being pulled) and watch the arm immediately move inward.  That's the inward component of the force Vector.

Now, imagine a stylus locked into the groove.  As it pulls the arm forward the arm pushes inward, pushing the stylus against the inside groove.

Remember, it's not the stylus pushing inward.... everyone seems to think it's the stylus generating some sidewards force.  No, the force of friction on the stylus is merely pulling it forward, then the arm reacts by exerting a force sideways on the stylus.  We need to exert a force on the tonearm to pull it outward.  That force = sin headshell angle x tracking force x coefficient of friction to perfectly balance the force.

If you do not balance this force by an equal and opposite force to the tonearm in the opposite direction the stylus will be pressing against the inner wall.  That's the reason for anti-skate....to equalize the force on both sides of the cartridge, which results in minimizing wear maximizing tracking ability. 

One way to balance the force (aren't there many ways?  Yes.) is by a very direct method.  You get your eye down as close as possible looking at the end of the cartridge.  Look at it square on, and if there is not a center mark right above the cantilever make on, on the front face.  Hold it just off the record surface with the cueing lever, then drop the lever and closely watch which way the cartridge body (the mark) moves relative to the cantilever.  It's not the stylus that moves (it's locked in a groove!) but the tonearm/cartridge, and it happens immediately.  Repeat over and over, lifting lever slightly, dropping, lifting, dropping, until you are confident you know which way the arm moves.  I just did this with an Orpheus as well as the Dynavector, so it's possible to get a good feel if you trust your eyes and memory.

If the arm moves in, add anti-skate.  If it moves out, reduce.  If it does not move, leave it.

To test your sensitivity to this test, move the weight in and out, and correlate your findings of cartridge movement with the weight changes.  You'll note a good correlation.

Try it with different records.  You will see that this does not change to any material degree from record to record, so you will know that the folks who say friction is a lot higher in a highly modulated groove are wrong.

Remember, frictional force is directly proportional to stylus pressure, so if you change tracking force you will have to change anti-skate.