Electronic – MC34063 with external FET. Why

At 100% duty cycle, you are correct in that the diode will not conduct. Also, your buck inductor will saturate out and you'll have the input voltage (minus resistive losses) applied directly to your LED with no other means of current limiting.

You're not driving the MOSFET correctly. Your pulse voltage should be from gate to source, not gate to output return. Buck converters with N-channel series MOSFETs need a high-side supply. The MOSFET will never be able to fully turn on with gate and drain close to the same potential with respect to source, since the minimum gate threshold for this part (with respect to source) is around 1.1V.

Tangent: you said you want a constant-current supply, yet in your powertrain you're not measuring the current that you wish to keep constant. I don't quite follow how this is supposed to work. I would expect that you have some sort of current sensing element (like a resistor) in series with your LED, which would be used to generate a voltage to control the duty cycle of the buck to make the current constant.

You have several things to worry about in this setup. A schematic of what you have exactly would have helped to target the advice more to the exact problem, but here goes:

First of all, you may not necessarily need insanely large capacitance, but you need capacitance with a low ESR (Equivalent Series Resistance). If your 470uF capacitor has a large resistance the regulation of the buck converter will improve and the peak current capacity of the capacitor will increase, solving the problem from two ends. That's not to say you may not need a higher value if you don't want a dip at all.

Secondly, you need a low ESR capacitor on the input of each converter stage as well, to make up for resistance in batteries and/or wiring, so the converter has a local store of energy to pull from when the output suddenly needs 1 or 2 ampere for the motor's cold start.

From the datasheet I expect the response of the regulator to be quite sharp, so it should be searched in the capacitors first. If you cannot get or afford a very low ESR capacitor, you can combine two or three lower value capacitors, as their ESR will be used in parallel and as such the total ESR of the capacitor-group will be much lower.

The second picture does seem to show a retry/restart of the motor and the regulator's response to that transient current. It's likely the servo has a little margin when the power to the motor falls off. More advanced servos are better tuned to prevent from showing behaviour like this.