Question is: Is something true or not? If you can prove it then you know it is true. If you cannot prove it all you know is you cannot prove it. It still could be true, or false. So the best you can do is to TRY to prove it, and/or try to prove its negation.

As pointed out by others generating random true statements is not very interesting because you can generate such trivially by appending && true as many times as you want to the statement.

So in general you always want to prove something, because only that can tell you something about what is true and what is not.