Clay is not an abrasive. It's just clay that pulls contaminants off the top layer of paint (things like sap and road tar).
A true polish will remove the oxidized layer of paint. You start with the most abrasive that fits the situation, and then work your way down to extremely fine polish to remove scratches from the previous product. You can get very fine automotive polishes that aren't so abrasive that they damage the paint; you'll see a step-by-step process in most makers' lines (Mother's, Meguiar's, etc.). Same sort of process of stepped polishes in products made for polishing plastics.
Most polishes nowadays have extremely fine abrasives because of the clear coat on almost all automotive surfaces, over the color coat (any metallic paint has a clear coat). "Polishing Compound" is NOT what you want; it's intended for really aggressive work on repainted or highly damaged paint, and it'll take off stuff very, very fast.
The best results on painted surfaces come from trying to remove the oxidized layer and remove scratches; only after that would you apply a glaze and then a cover coat of wax. The "all-in-one" products are compromises for those who are lazy.
Another thing you could try is Flitz polish, a non-abrasive metal polish also used for fiberglass and plastics.
As noted, much depends upon what kind of paint you have. And how much color coat is really there -- multiple layers, or just one thin one. Obviously, if you're removing oxidation, you need to have sufficient good paint underneath!
Any polishing should be done with either a microfiber cloth made for that purpose, or 100% pure cotton flannel, tossing the contaminated cloth as the polish does the job.
If you care about the finish of your bike, never use any scouring pads, nylon brushes, or the like to wash it! Same with a car! Use only microfiber products made for that purpose, or a real boar bristle brush with flagged tips.