That's a fallacy and you know it. It's only a problem to the self-debilitating sceptic. Peeps make careers out of such irrationalism.
You can't disprove that there's a flying teapot orbiting the Earth, exactly halfway between Us and the Moon.
You can't disprove ghosts, fairies at the bottom of the garden, tooth fairies, and all other superstitions. But that doesn't mean the 'Yes' and the 'No' weigh the same amount.
Exactly. When you claim something exists or has happened, the burden is on you to prove it, not on a skeptic to disprove it. If a religious person claims there is a God and an atheist says there isn't, the burden is on the religious person (at least if the goal is to ascertain the truth as far as is possible).
The only exception to this is when the person affirming something has already been judged to have satisfactorily proven their case--then the burden shifts to the person saying it's not true or didn't happen. For example, if someone is convicted of a crime and there were no legal flaws in the trial (admission of tainted evidence, for example), if the defendant still claims to be innocent the burden is now on them to prove they didn't commit the crime.
