As far as we know, the natural world is all there is. If there are realms that we cannot know, then there is no use in speculating upon them. Weak naturalism limits itself to what we know. Just as a weak atheist simply disbelieves in God given the lack of evidence, weak naturalism disavows the supernatural for the same reason.
In only asserting the existence of the natural world, the burden of proof is transferred to the “supernaturalist.” Proposing weak naturalism does not require positive evidence showing why it’s probable that nothing transcends nature. Rather, it appeals to the lack of evidence for anything supernatural, period. It’s not scientism to expect knowledge-claims to be verifiable or testable. The scientific method has become the accepted method for ascertaining which empirical claims are true or not for a reason.