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That does actually exist, it's called a toroidal transformer. Instead of a tube shape though it's a donut shaped piece of ferrite with the primary and secondary wound one on top of the other. The different types have different use cases though, as far as I remember the toroidal ones have higher leakage current. Marco Reps has a good video on his YouTube channel where he explains some of the differences if you're interested, although he mainly focuses on what's important for precision electronics.

1. Insulation. Physically separating the two sides makes sure there are no issues.
2. The difference in magnetic fields distance is fairly irrelevant with a core that is really good at "conducting" magnetic fields.

I'm not sure I understand your proposed solution, but the ferrite core guides the magnetic flux, there are very few losses, and both circuits are perfectly electrically insulated from each other. How would your solution be an improvement?

The coils in EI-core transformers are all wound around the middle leg. My last build had five separated coils: primary, 520V, 80V, 25V, 6,5V.

This is how a toroidal transformer is built. https://en.wikipedia.org/wiki/Toroidal_inductors_and_transformers

It's all about the lines bro, the lines.