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First of all, your example doesn’t show any «instantaneous travel». You can shuffle two playing cards, send them by light years. Flip one of them and instantaneously determine what on the other one.

There are more complicated correlations in QM than cannot be explained by hidden variables - but those are still correlations and, analogously, don’t imply any «instantaneous travel»

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Flipping one card doesn’t change the state of the other one. Measuring the state of one of a pair of entangled particles does change the other one & that change appears to be instantaneous. We know this - Aspect carried out the experiment in the early 80s & those results have been confirmed by later, larger scale experiments.

(IIRC It’s possible that a super-deterministic hidden variable system could explain Aspect’s results. But that pretty much requires the universe to deliberately plot against you, which seems very unlikely.)

> Flipping one card doesn’t change the state of the other one

Right, but you didn't say "change" you said "determine".

> Aspect carried out the experiment in the early 80s

I have a write up of the maths behind this if anyone is interested, accessible to an undergraduate: https://h2.jaguarpaw.co.uk/posts/bells-theorem-made-simpler/

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