What am I missing here when it comes to entanglement and the Stern-Gerlach Experiment

So I know there's been a ton of posts regarding this lately due to a certain content creator that released a new video that involves it. I apologize for contributing to the flood of posts on the subject- I've asked my question in several more general "question" threads and haven't found the answer, so I'm going to see if posting it as its own thread might help. I've been wondering about this since I started goinging through Prof Allen Adam's wonderful Quantum Mechanics lectures on MITOpenCourseware's Youtube channel, and I really thought I understood this, but I've got to be missing something.

In the first lecture, he gives an example of a setup where a beam of electrons goes through a device that splits X-axis spin up from spin down, then up output from that goes into another device that splits Y-axis spin up from spin down, and then up output from that goes into a third device that splits across the X-axis again. He goes through how you expect them all to come out spin up, because before the Y-axis splitter, they were all coming from the spin-up-output of the first X-axis splitter. But instead, they come out 50/50.

Now, my understanding is that of the original 100%, 50% come out X-spin-up from the first splitter, then 25% of the total come out from the Y-spin-up output from that splitter into the third splitter, with 12.5% of the total coming out X-up, and another 12.5% coming out X-down.

That means of all the original electrons, 62.5% wind up being X-spin-up. I imagine you could further extend the setup to get as high of a ratio finishing X-spin-up as possible by repeating this process.

This doesn't seem right to me, because my understanding is that if its done with entangled particles, you could put them through this sort of set up to change the ratio of the partner particles being measured X-up/X-down. Which I know isn't allowed, so what am I missing here?