I am guessing that there may be a fundamental physics limitation limiting the output current given such a low input voltage. Any thoughts about that?
No. It is irellevant. Most MOSfets are enhancement types, and they need some applied voltage to open, but SMPS circuits for low voltages often have a built in charge pump to generate a high enough gate voltage for the FET.
(Are Depletion MOSfets even made these days?)
What about rolling my own
There are some very simple "joule thief" circuits but they rely on hand wound inductors, have to be tuned individually and work only reasonably well in a small power range.
Rolling your own can be a very good learning expericence, but if you will be able to make a well behaving circuit is not ensured. A starting voltage of 0.8V is a bit difficult to achieve. Adding a charge pump may be necessary (increases complexity), or if you can get it started, you can use the output voltage to have a high enough voltage to switch the FET's.
Controlling all the things is also a brain teaser. There are some microcontrollers that can work on 800mV I would probably start there. You can use it to generate PWM signals, put the control loop in it, and maybe a voltage divider on a digital pin is good enough to get a stable enough output voltage. You definitely want a synchronous converter with such a low input voltage, as even a single diode (Even schottky) will have a high enough voltage drop to get the efficiency below 50%.
Also keep in mind that if you want 3V7 @200mA at the output and your input is 800mV, then you need 925mA on average at the imput. Peak current during PWM is likely more then 2A. But SOT-23 FET's that can do 5A are readily and cheaply available these days.