everything simply becomes belief and opinion and there is no more truth
This is really striking! I think truths remain as truths until they're disproven. But I think you're right about beliefs. We can't do away with them because we have to make sense of our world no matter what and that means clinging to any form of belief that we can.
It's the other way around. The burden of proof is on the claim being made. If you make a claim of truth, the burden lies with that claim.