If you intend to be sure that rejecting the Null Hypotheses it is almost surely false, YES, you use very small alpha. Why it seldom the possibility is used? Simply because the type II became inadmissibly large: whatever the Alternative Distribution you will fail to reject it . . . Saying in passing that the Ulrich´s use to say (at least since 2004): nothing is normal is a worthless truism. In fact there is no way to be sure about the Distribution a sample is drawn, Normal or whatever. In NHST we do not try, because we cannot, really to find out the Distribution, but, if the result is not rejecting H0, we conclude that nothing is against to admit it and proceed as it was true. By the other hand the risk to reject a really normal sample is controlled by alpha. All GoF (Goodness-of-Fit) tests likewise. The Ulrich´s confusion seems to be chronic among unlearned practical statisticians/users: the uneasy they felt from a structural no-certitude of all random procedures leads them to say that based on a H0: p=0 no rejection we should conclude by the parameter´s nullity. Abuse, of course, one only conclusion is valid: there is no evidence that p is not 0, which is rather different that to state p=0. The same for GoF: the real problem is how a no-rejected normality test sample behaves before an ulterior more or less *robust* test. . . As you surely got NHST are not fair in what concerns the two Hypotheses, H0 and Ha, it favors the former. It Is better not to claim difference, even if it really exists, than to discover them when they are merely random fluctuations. Sure?
One advice: Read what David Jones said: He is absolutely right.