Double_R
But what are their sides? I would argue your last paragraph is fine.

Let's imagine that. Pro presents 3 arguments to prove X. Con successfully refutes all 3 arguments. So what are we left with?

If you started this debate in the Con position, by the end you're still in that position. But of you started in the Pro position, then having no arguments left to support it, you're no longer rationally justified in that position.

So in short; Con = unchanged, Pro = changed. That's clearly a Con victory.
Double_R
The BoP more than just a debate or legal concept, it's a basic principal that we all use to govern rational thought. If I told you aliens were going to wipe out planet earth in 3 days, would you call out from work and start dipping into your savings since you're not going to get to use it otherwise? No, of course not. The time to believe something is when there is evidence for it, so the proper response to me would be "prove it". And in my inability to do so, you have no obligation to change anything.

When you want others to move from their position towards yours, whether that be in debate, in court, or anywhere else, you have to do the work. That's the idea.
Double_R
I think you're conflating the burden of proof and the standard of evidence. The burden is just the question of which side is responsible for making a positive case. The standard is what addresses the strength of the case required to win.

The BoP doesn't require one side to convince the judges while the other gets to tweedle their thumbs, it just means thier responsibility is to argue for the resolution while the other is responsible for negating all of their arguments. If the challenger decides not to make a positive case for the alternative, then they're putting all of their onus on refuting each point made, otherwise they lose. That's how I see it anyway.