Regarding that last point: You may know where you left your keys, even if you are "only", say, 99% certain where they are. Now, how strongly do mathematicians believe that the Riemann hypothesis is true? 99.9% perhaps? In some sense then, you already consider yourself less ignorant about the truth value of RH than about the location of your keys, or about many other mundane things you know without being perfectly certain about them.

A proof would then merely update you from 99.9% to ~100%, which is a smaller update than the example of checking that your keys are indeed in your pocket, where you go from 99% to ~100%.