Output Transformers, why are they made like this?

[Kalee20]

Which is why I considered winding the primary on the outside, away from the core.  The potential difference between the core and the secondary will be negligible.

 I'm going to try it at my next winding session.

Do! It'll still work.

But, granted that the pd between secondary and core will be negligible... you've still got the pd between primary and secondary! And that won't go away.

(Connect it so that the primary start goes to HT+ and the outermost of the winding, in free air, goes to the anode, so that the high peak voltages are well away from everything).