Many years ago we did build the circuit shown in the app note. Like yourself, we also attempted to use Mosfets instead of SCRs, but after a pair of attempts, went back to SCRs.
Why? Very simple: SCRs self commutation. They turn off once the current drops below the holding current. One only requires to trigger at the correct phase angle and the self commutation does the rest. Mosfets, on the other hand, requires a turn off signal synchronized with the current. Because the current’s zero cross varies with the load, one must sample the pulsing current.
Another failsafe SCR feature is that it cannot conduct reverse current, even if incorrectly triggered. A Mosfet which is accidentally turned on while the other is conducting (for whatever the reason, s**t happens) will result in a dead short across the transformer’s secondary winding.
Another SCR advantage is that you trigger it and forget, the Mosfet OTOH, requires continuous high side drive for the duration of the on period.
The result is that you require a more complex drive circuit. It can be done, but is more complex.
Now, the EMI is practically the same. The dI/dt is limited by the inductor. Whatever voltage ringing, can be quenched with a snubber.
The SCR has an additional advantage: for a particular die size and cost, the SCR’s fusing (I2t) capabilities are more rugged.
Most of the battery chargers we built 28 years ago are still operating in industrial environments.
Now, if you want to still use Mosfets, be my guest.