The burden of proof tennis is quite tricky here because it’s not about whether you claim something exists, it’s whether you claim something that goes against what’s generally accepted. If I claim quantum mechanics don’t exist, it’s not on you to prove they do.
And that’s before we get into the fact that there isn’t a general consensus on whether God (or any gods) exist.
It was on them until society generally accepted it. Now if I claim it doesn’t exist, the burden is on me.
Or how about this: if I claim dinosaurs never existed and thus the fossils didn’t come from them, it’s not on you to prove they did.