R3 forces white to choose between playing R2 and S3. If white plays S3, white can just capture like you say. But P1 becomes sente for black. If white plays R2 and white extends at S3, white can still play S2 to kill the stones, but then S8 is sente.
Basically R3 makes white choose, and black can have the resulting sente later.
There are big endgame moves at P1 and S8. However each is gote. The cut sacrifices a few stones at no cost (unless it makes W alive when W wasn't), for a larger threat, not just a boundary play but also threatening the entire W group. It turns out that whichever way W responds, either P1 or S8 becomes sente. So it maximises the profit.