Majority opinion does not make something the truth.
If you need to appeal to majority opinion rather than actual evidence and arguments, you really have no case.
The burden of proof relies upon those making a claim.
If someone claims Earth is round, the burden of proof is on them to show it is.
If someone claims Earth is flat, the burden of proof is on them to show it is.
A REer claiming Earth is round and failing to meet their burden of proof, does not mean that Earth is not round, or that Earth is flat, just that they have not met their burden of proof.
Likewise a FEer claiming Earth is flat and failing to meet their burden of proof does not mean that Earth is not flat, or that Earth is round, just that they have not met their burden of proof.
This ties in quite nicely with theist.
If a theist claims their god exists, the burden of proof is on them to show it does. If they cannot, then it is fine for people to reject their claim as unsubstantiated. An atheist is under no obligation to prove the nonexistence of all possible gods to justify their lack of belief in a god.
However if the atheist goes further and asserts that gods do not exist, then the burden of proof is on them to show they don't, and it is fine for a theist to reject that claim of non-existence, but not to asset that a god exists.
So both sides have a burden of proof, regardless of what the majority believes.
One side not meeting it doesn't make the other side correct, nor does it mean they don't have one.