... "aren't SE transformers subject to saturation at lower frequencies, and therefore, SE transformers that handle lower frequencies larger by design ...
ALL transformers have a low-frequency limit imposed by the size of their cores, so to go lower they need to be bigger. This is about
power through-put, so if you reduce the applied power by an order of magnitude, the low end extension increases by an order of magnitude. So it's really a power-bandwidth product.
What is being used up is the ability of the core to handle/transfer magnetic force created by the power applied to the primary winding.
... and therefore, SE transformers ... require air gap design considerations?
All SE transformers use up a portion of the core's magnetic capability in the unbalanced direct current through their primary, which is the output tube's idle current. The air gap helps matters, but doesn't eliminate the unbalanced d.c. issue. OT manufacturers usually specify a maximum unbalanced d.c. for their SE transformers.
The follow up, doesn't a transformer that is saturated gets hotter?
Not necessarily. Or at least, not due to the saturation.
Heat is caused by I
2R losses, also known as copper loss. "I
2R" means "current
2 times resistance" (of the copper winding), which is one variant of the equation for power. So high current through a high resistance winding implies large power dissipated as heat (which is not usually an issue in typical guitar amp transformers).
Eddy currents in the core can result in heating of the core, but that is why the cores are usually made of laminated steel plates.
So really, core saturation will cause the transformer to be unable to pass more power through it, or to have its bandwidth restricted, but doesn't necessarily create heat without some other fault occurring at the same time.