PhD in maths here, no longer a mathematician, partly because I wanted to do stuff more connected with real life.
I actually want to see interesting theories and mathematical ideas come out of proofs more than I want solutions/proofs. After all most maths doesn't have direct practical applications. So although unsolved problems are a good barometer of 'there's still stuff left to understand here', the interesting part about solving them is less knowing what is true and more 'how does this help us understand this field better than we did before?'.
I would still like to know if the Riemann Hypothesis is true though!
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%.