How do Many-world interpretation explain we live in a "statistically right" universe?

The answer here addresses a common misconception about statistics and reality. If you’re asking why we live in a universe where the odds when flipping a coin are 50/50, remember that the universe is of finite size, so there are finitely many coins to be flipped. There’s a good change the actual prevalence of heads vs. tails as far as spin states are concerned is not exactly 50/50.

Yet the law of large numbers guarantees that the difference will fall within a measurement error of 50/50, because across the normally distributed set of all possible parities for this spin state coin flip value, the odds of deviating even by the smallest measureable error are very, very, very low. Astronomically low, you could say :)

The first problem with this question is that it is somewhat like the Inverse gambler's fallacy.

It assumes that in a universe where things have frequently been "wrong" things will continue to go wrong. In any such universe, the chance of something going wrong in the future is the same as the chance of something going wrong in our own "perfect" universe. When it comes to statistics, history is irrelevant.

The second problem is that the question assumes "that the statistics is always right" in our universe.

Happenings that some people interpret as miracles, one-time-only experimental results, and various other forms of "impossible" events are generally dismissed by the scientific community, even if only because they can't be reproduced.

If a massive statistical failure ever should occur, it is unlikely to be observed. And if it is observed, the observer is unlikely to believe it actually happened. And if the observer does believe it and reports it to others, no one else is going to believe it.

It doesn't explain it. Neither does any interpretation of quantum mechanics that reduces quantum amplitudes to classical probabilities.

Suppose I claim that a coin comes up heads with some classical probability $pin(0,1)$. What does that mean, operationally? In other words, how can you test it? You can flip the coin $n$ times, and if I'm right you should get $np$ heads. Actually, that's not true, you could get any number of heads. Well, at least certain numbers of heads are more probable than others, assuming my claim is true. You can prove that if my claim is true then if you flip the coin $n$ times then you'll get between $k$ and $ell$ heads with probability $q$, where $q$ is a complicated function of $p$, $n$, $k$ and $ell$. The trouble is that this new probabilistic claim is on exactly the same footing as the original one, and if you try to give an operational meaning to it, you get an infinite regress.

In practice, we deal with this problem by inventing a cutoff $ε>0$ and reasoning as though anything with a probability less than $ε$ is certain to not happen. But there's no way to formalize this idea without destroying the internal consistency of the probability calculus.

The problem in many-worlds is exactly the same, and can be "solved" in exactly the same way, by inventing an $ε>0$ and deciding that worlds with amplitudes within $ε$ of $0$ don't really exist. This isn't a solution at all, but neither is adding the Born rule to the theory. That merely substitutes the classical-probabilistic version of the problem for the quantum-probabilistic version.

Deterministic theories like Bohmian mechanics have essentially the same problem. Among all the possible worlds (given by different initial conditions), there's no obvious reason why the actual world should be one in which experiments point us to the correct laws of nature.

This question is actually easier than many related questions about statistics, because this flavor of MWI lends itself to frequentist interpretations of probability. According to MWI (as you seem to be using it), there exist $2^n$ universes total after the measurement you describe. Of these exactly one universe has all copies of spin pointing up, and exactly one has all copies down. There are $2^n$ copies of you, but only $2$ of them see all spins the same. The vast majority of "you"s will see about half the spins up and about half down.

Notice that we do not live in a universe in which the statistics are "always right". We live in a universe in which the statistics are "right" exactly as often as you would expect based on the argument above. If you run the experiment described repeatedly, about $2$ in every $2^n$ runs will come out "wrong".

Extend your assumption to the entire visible universe. As others have pointed out, there will be at least two universes (a few more I would say, but still a minimal fraction of the total) in which the statistical results are such that quantum mechanics as we know it do not represent the laws of physics, even if they were true. Observers in such universe would come up with different statistical laws to describe what they observe. But of course, these "universes" would be outliers. So we live in a typical universe, I would say. Or in one of the outliers perhaps? there is no way to know.