Why is flyback air gap needed for energy storage?
Unlike a forward-topology transformer (where the primary and secondary windings are conducting at the same time), the flyback transformer must store energy during the primary switch on-time, delivering it to the load during the primary switch off-time.

A forward-topology transformer doesn't need any gap since the peak flux density is a function of the applied volt-seconds only; the power being delivered 'through' the transformer isn't a variable (other than its effect on duty cycle). It's only the magnetizing current that moves the core along its hysteresis loop, which doesn't pose any saturation risk if everything is well-designed, since the primary and secondary ampere-turns cancel each other out.

A flyback transformer doesn't have the ampere-turn cancellation benefit of a forward converter, so the entire \$ \frac{1}{2}LI^2\$ primary energy moves the core up its hysteresis curve. The air gap flattens the hysteresis curve and allows more energy handling by decreasing the permeability of the core. You will of course need to add more turns to get your desired inductance compared to no-gap, but you avoid core saturation.