Based on this logic no burden of proof exists for anyone.
No, there are burdens of proof all over the place. It's just that a chain is only as strong as its weakest link.
The claim "X exists" could be restated as "it must be the case that God exists." Unless every element in the argument supporting that claim is rock solid, we can only get to "it MIGHT be the case that God exists, but it might not"... which is where we were before the argument was even attempted.
If we're considering some element of the argument and I say "I don't know whether this is true", this implies that I don't know whether the claim being argued is true.
Basically, the person making that original argument can never get into a position where he's automatically right because of something his debate opponent said or did, or failed to say or do. Even if I respond to my opponent's argument inappropriately or in a way that somehow violates social expectations, his argument is still his to support on its own merits.
You are skipping over the fact that any conversation that would have a burden of proof is necessarily in the pursuit of truth. This is from where the obligation to respond comes.
If I'm interested in pursuing truth, it may often be the case that there are better ways of achieving that goal than engaging with every yahoo who decides to argue with me. Maybe I'd be better off abandoning the argument and finding a book to read.
But if we forgo that, then no one has any burden of proof. We are all just bumping around sharing feelings.
No, there's one burden of proof: the original claim.
The person responding to that argument might make other claims in response. Yes, these do have their own burden, and we ought not to accept them as true until their burden has been met. However, even a claim whose burden is unmet is a challenge to the argument.
I don't see why this is an issue. It's common - and IMO good practice - to question an argument by asking "have you considered X, Y, and Z? If any of them are true, then your argument fails." This doesn't mean I now have to go out and see whether X, Y, and Z are true or false.. and often I can't check them. But the mere fact that doubt has been raised means that part of the burden for the claim being argued is to either show that X, Y, and Z are actually true, or that the claim can be true even if they're false.