Consider now a similar theory, but multiple copies of you are allowed. The theory predicts that there will be one billion branchings of the world in the next second, with each branch containing a person who shares all your memories up to that point. The theory also predicts, as above, that all but one of these worlds will be obviously bizarre. As a matter of fact, as you read these words, you do not experience the world around you suddenly becoming bizarre. But unlike the previous example, this is entirely consistent with the theory, which predicted that one version of you would continue in the world as per usual.
Yes, your theory states that the chances are 100% that some copy will find itself in the non-bizarre world. But the theory also states that the chances are very low - one in a billion - that *I* will be that copy. Why isn't this second probability important? It seems to me that you only care about the first probability, and disregard the second as irrelevant.
I'm assuming that the hypothetical person in these thought experiments is not certain that the copying theory is actually correct, and is trying to figure out whether or not to believe it. In such a case, the one-in-a-billion probability is relevant. If the theory states that a billion copies of me are going to be made, and only one of those copies will find itself in a non-bizarre world, and then I find myself in the non-bizarre world, this should change my assessment of the likelihood of the copying theory - it should make me a billion times more suspicious! Whether or not this causes me to reject the copying theory depends on how much credence I gave that theory before the alleged copying took place. If I thought that the copying theory was almost definitely true - say, a trillion times more likely to be true than false - then I should continue to believe it after finding myself in the (very unlikely) non-bizarre world. If I thought that the theory was only a thousand times more likely of being true than false before the alleged copying, then I should strongly doubt the theory after I find myself in the non-bizarre world.
If we accept your logic, the Platonia idea is unfalsifiable. One could, I suppose, still believe it on strictly a priori logical grounds. Is that your position?