First, atheism is a response to theistic claims, but that does NOT necessarily make it the opposite of those claims, nor an active disbelief in those claims. More often it is a lack of belief in the claim rather than an active belief in the opposite of that claim.
For example, if I said I had a quarter in my pocket, but I wasn't going to show it to you. You could either believe it, actively disbelieve it, or say that you will neither believe nor disbelieve until evidence is given. You reject the claim that I have a quarter in my pocket until proof is provided, but that does not necessarily mean that you actively believe that I do not have a quarter in my pocket.
That said...burden of proof.
Yes, burden of proof IS asymmetrical...as it just about ALWAYS is. Heck, if it were not asymmetrical, then there would be no "burden," would there?
But there is a burden and it is always on the party making the positive claim. And the more extraordinary the claim, the more extraordinary the proof required to meet the burden.
In court, we call this innocent until proven guilty; that is to say that the prosecution must prove its positive claim that the defendant committed a crime and until they do so they defendant is to be considered innocent of the charge. This is just one example. But there are many.
For example, if I say that Bigfoot is real, is the burden of proof on me to prove that he is real or on those saying he is not? The rational stance is to say that Bigfoot is not real until it is proven that he is and the burden of proof is on the person saying he is. Substitute any number of things for Bigfoot and the logic holds--fairies, the Loch Ness monster, UFOs, ESP, god, etc.
The burden of proof remains on the person making the positive claim and the rational position is to not believe the claim until it is proven true
Now, if someone says "I know there is no god," then maybe you could say they are making a positive claim in the negative direction. But if someone says "I do not believe your claim until you prove it," they are making no claim of their own and hold absolutely no burden of proof.
Now, you claim that proof is already given. Where? What form? I've never seen any, not empirical proof anyway. I've heard anecdotes and unconfirmable personal accounts, but no one has ever produced any actual, provable, verifiable, concrete, empirical proof for any gods' existence. None. If you think otherwise, please tell me exactly what proof you are referring to. Empirical proof only, please.
Finally, a lack of belief is not and cannot be defined as "an ideology." Is it an ideology not to believe in vampires? Is it an ideology to not think Stephen King is that great a writer? But most importantly, an ideology is, by definition, "a systematic body of concepts." Atheism is a singular stance on a singular claim - the rational rejection of a god or gods until evidence is given otherwise. You cannot, by definition, have a single-issue ideology. It simply does not qualify. And as there are NO other central, universal claims, morals, tenets, or beliefs shared by atheists, atheism falls far short of qualifying as an "ideology."