Regulating power on a Peltier module
You do really want a switching regulator, whether it's a commercially bought 'switching regulator', or the 6A 'alternative circuit' that you link to, which is PWM with output filtering.

As explained in the article around your alternative circuit, straight PWM, though efficient for the supply, is not efficient for the Peltier. As the cooling power depends on the current, but the resistive losses depend on current squared, it will generate more heat, that is run at lower overall efficiency, if there is any fluctuation in its supply. The ripple of a switch mode regulator is just fine. Slamming the power between 0% and 100% with PWM is not.

To look at some other PWM/switcher applications for comparison ...

a) you can PWM a filament lamp, as the heat output is what you want
b) you can PWM a brushed DC motor, stepper motor or solenoid, as its intrinsic inductance smooths the current out
c) you shouldn't PWM a Peltier, or a charging battery, because you want the current, and the excess \$I^2R\$ heat that you get during the 100% on phase is unwanted. You need more or less smooth DC for these applications.