Although I agree with Allure's answer (obviously, otherwise I would have to reject Bell's theorem!), at the risk of going off topic (in which case downvotes will let me know), I just wanted to make a comment on physics, models, observations and the operational approach, particularly about this statement in your question:
But measurements apart, does it explain anything about how nature works, in general?
Remember that the goal of physics is to make models which explain observations. No matter how hard you think about how the universe actually works, whatever that means, that is a question that's better left to philosophers. The only things we can talk about is stuff we can measure, physically, in a lab.
And to an extent, this is not a bad way to define "how the universe works": this is the operational approach. To put it in a harsh way, if I can't distinguish two models with experiments, they're the same, and none is better or closer to the truth than the other. In a sense, they're both the truth, as the true workings of the universe are ill defined unless we reference experiments. There is no a priori way to describe the universe as we are fundamentally observers.
Let's now get back to the uncertainty principle. People will tell you something along the lines of:
Position and momentum of a particle truly aren't defined simultaneously because $\hat x$ and $\hat p$ do not commute, and noncommuting observables share no eigenstates.
this is true, but when you hear this keep in mind that $\hat x$ and $\hat p$ are nothing but a model for our observations. At its core, the uncertainty principle is only about measurements! We can build a model in which the statistics of our measurements are calculated in a different way, using something called "hidden variables" that sit closer to our intuitive understanding of how the universe should work, but it turns out that these two models are distinguishable experimentally, and John Bell proved so. So people set out and did the experiment, and the uncertainty principle won. But keep in mind that hidden variables theories still talk about a hidden variables influencing measurement statistics.
In this light, what the uncertainty principle and Bell's theorem tell us is that experimentally we can never know both the position and momentum of a particle exactly at the same time, and there isn't anything that, if we could measure it, would help us gather this knowledge (a hidden variable).
Whether or not this means that the particle doesn't really have a position or a momentum, or even if the particle exists at all in any way that our human mind can conceive, is a question that according to the operationalist is outside the domain of physics.