On a driven sharpening device (like the wheels), softer abrasives can sometimes work well, but only because of the velocity at which the polishing action occurs. On a strop by hand, not so much. Most toothpastes use what's called 'hydrated silica' for their abrasive, and it's not very aggressive stuff on hardened steel. Think about it; if toothpaste were aggressive enough to polish hardened knife steel by hand, would you really want to be using it on your teeth? It'd strip the enamel away from your teeth in short order.
On a strop, I wouldn't waste time with the toothpaste, as it's likely not doing any more than a bare leather strop would do. If you want to see how much it works, apply it to a piece of clean, white paper and strop on it. The lack of any grey/black streaks left on the paper will tell you how minimal the real polishing (metal removal) is. It may serve more as a 'cleaner', removing debris and other loose contamination from an edge, but will do little more than that.
David