The purpose of the brain-in-a-vat scenario is not to convince anyone that we might really be a brain in a vat; the purpose is to illustrate the fundamental disconnection between perception and reality. Before the brain-in-a-vat scenario, Descartes made the same point by discussing dreams.
The classical view of perception is that when you, for example, look at a tree, you are seeing the tree; you are experiencing the tree; the tree has become present to your mind. But as the brain-in-a-vat or dreams show, you can have exactly the same visual experience of the tree even when there is no tree there. What this means is that there is some disconnect between what is really out there (which we might call the noumenon) and what is perceived (what we might call the phenomenon). Arguably then, you are not experiencing the tree; the tree is not present to your mind; rather, you are experiencing a mental image; it is the image that is present to your mind, not the tree itself.
Apparently, there is a sort of chain of causality like this:
tree --> light --> eye --> ... --> [X factor] --> experience of a tree,
and another chain of causality like this:
[something that is not a tree] --> [X factor] --> experience of a tree.
The experience of a tree is something that is somehow constructed by the [X factor], not by the tree itself. One could assume that there are two entirely different mechanisms, one that transfers the image of a tree into mind and another that constructs the image of a tree, but there doesn't seem to be a good reason to assume this.
So, if the experience of the tree is something that is manufactured in the brain (or the mind) by [X factor], then we have a real problem: what reason is there to think that the tree is anything at all like the way we experience it? In other words, the image of the tree that I have in my mind is a useful way to interact with the world, but what reason do I have to think that the noumenal tree is anything at all like the phenomenal tree? If I am a brain in a vat, the noumenal tree might just be a data structure in a computer program. If I am dreaming, the tree may be a bit of undigested beef, a blot of mustard, a crumb of cheese, a fragment of an underdone potato (in the words of Ebeneezer Scrooge). Physics even backs this up. According to physics, the tree is really just a swarm of subatomic particles that our brain presents to us as a complete unified tree.
So, the answer is, no, the brain-in-a-vat scenario does not shift the burden of proof, because the point is not to prove that we are (or might be) brains in a vat; it is just a thought experiment intended to show that we have no reason to think that the noumenal world is really what it appears to be. An idealist would say that there is no noumenal world at all; there are only the experiences. A Kantian would say that although the noumenal world may exist, we can't know anything about it through perception; we can only know about the phenomenal world.
I should note that there are a few writers who are trying to prove that the universe is a simulation, but these are mostly physicists and computer scientists, not philosophers, and they claim to have evidence beyond the brain-in-a-vat scenario, but as far as I know, almost no one takes them seriously.