“On the nose” didn’t come from horse racing, it came from radio.
Several common hand gestures came from the early days of radio broadcasting, when elaborate productions required the director in the studio to be able to communicate without speech, and so they used hand signals.
For “cut” a forefinger was slashed across the throat.
Holding up the forefinger touching the thumb meant “good performance,” and touching the nose signalled “perfect timing.”
It was right on the nose.