Why are golf balls the only (famous) object with dimples?

The key is that dimples are not magical. Their purpose is to drag the boundary layer with the ball longer. If it can do this, it can reduce drag. However:

People have thought of putting dimples on everything from swimsuits to cars to airplanes. You only get an advantage from these dimples if the boundary layer can be made to stick longer to the object. Some cars just have vertical flat ends to them where the trunk comes down and there is no way to reduce the turbulent wake of these no matter how dimpled the paint is. And the boundary layer stays with airplane wings except maybe a bit at the ends (some gain can be made by putting small rods out on the tips or on the trailing edges of the wings).

So the key limitation is the geometries which benefit from dimples. In many cases, the shape of the object can be adjusted to be more aerodynamic, reducing the need for such dimples. In other cases, the shape of the thing which is needed to cause this effect isn't a dimple. It might be a wire or a spoiler.

Which leaves spherical projectiles, which is where your question focuses. The real question is, how many spherical projectiles are there in which we care about their performance? Cannonballs went the way of the dodo a century ago. There's simply not all that many spherical projectiles out there.

Except in sports. In sports, we find all sorts of spheres, and you'll find all sorts of rules about them. In baseball the answer is simple. If you get caught intentionally scuffing the ball, you get in trouble and the ball gets replaced. The game is balanced around an un-scuffed ball. In cricket, scuffing the ball is actually part of the game mechanics. You'll see players intentionally scuffing one side of the ball to create an uneven effect that permits the bowler to have more control over the ball.

In shot putt, I think they giggle at the idea of air friction against a 16 pound projectile, so dimples aren't needed.

It's the combination of shape, speed and size that makes dimples advantageous in golf balls but not much else.

It all comes down to the flow condition at the point where flow separation occurs. In calm air, every boundary layer starts as a laminar boundary layer. Since the energy transfer across the stratified flow in the laminar boundary layer is reduced to shear, the molecules close to the wall will lose speed quickly, so that even at a modest pressure rise downstream separation occurs quickly.

Inside the laminar boundary layer, small disturbances become less and less damped the higher the local Reynolds number becomes, and at a Reynolds number of around 400,000 in unaccelerated flow some frequencies become unstable (see Tollmien-Schlichting waves) and will eventually create so much cross movement that the boundary layer becomes turbulent. Now parcels of air which flow at high speed in the outer part of the boundary layer will move close to the wall and kick the slow parcels there ahead, greatly reducing the deceleration of the flow close to the wall, at the price of slowing down and expanding the whole boundary layer.

Such a turbulent boundary layer is much better in following a contour with an adverse pressure gradient since it shows less deceleration of the flow at the wall. A pressure rise is generally caused by a contracting body shape. Separation is delayed and the separation, once it occurs, is much smaller. The pressure in separated flow is lower than ambient, so rearward-facing areas with separated flow cause massive drag. Therefore, separation needs to be suppressed as long as possible to minimize drag.

If the contour of a body contracts at a local Reynolds number below that where natural transition to a turbulent boundary layer occurs, the still laminar boundary layer will cause early separation. The dimples of a golf ball help to trip the boundary layer early into its turbulent version, thus delaying separation and reducing drag.

If the local Reynolds number (which is proportional to the product of speed and body length) is higher, such that the boundary layer turns turbulent before the part of the body with the adverse pressure gradient is reached, dimpling the surface will still cause earlier transition, but will not change the separation location. Faster and/or larger objects than golf balls simply do not need dimples.