Universal generalisations are certainly testable, even if not provable, in the sense that it is always possible that the experiments we perform or the observations we make should turn out to falsify them. So the substitution of testability for provability allows universal generalisations to be included in science all right. Indeed Karl Popper has built a whole philosophy of science on the principle that what distinguishes science from non-science is its ‘falsifiability’.
This weakening of the empiricist requirements on science does not really solve the problem of induction. Even if the requirement of testability succeeds in picking out what people standardly and intuitively count as proper science, it leaves us with a problem of explaining why such proper science is a good thing. We have still been given no account of why success in past tests should be a good basis for accepting generalisations which predict the future.