So, the first thing to test is a piece of wire. Yes, crazy, but it makes it easier to debug.
Here is the scope view:
The yellow trace shows when the high transistor is on (briefly to add a bit of current to the inductor). Then, it shows that the low transistor is on for a long time (it takes a long time to reduce the inductor current). The inductor current is middle of the scope and shows the fast rise when the high transistor is on, and the long decay when the low transistor is one. The voltage is purple, and just ripples of voltage (remember, the is just driving a wire).
Next, I connected a supercapacitor, and started the charging.
Here is the scope view...
So, it worked! It keeps the inductor current hovering around 1.2V (i.e., the 1.2V of the comparator), and the capacitor voltage is slowly increasing.
Total capacitance, 450 F, total charge time from 0.7V to 2.8V, 2 minutes, 53 seconds. So with the untuned controller, the average current was around 5.5 Amps. Next step would be adjust the controller code to tighten up the current ripple (it gets a little sloppy when the capacitor voltage increases past 1 V).
In still air, the hottest component on the board was the low transistor at 50C.