"Enable Switch"?
Looks like a PolyFuse. "3" could be amperes?
You can trace the circuit if you want. Guessing from the 555, it's embarrassingly bad, which seems supported by your observations (lack of output regulation, sensitive control). If I had to guess, I'd bet the trimmer is duty cycle, so, roughly it controls energy into the winding (roughly a flyback topology, but the CW multiplier complicates that), and output will be load sensitive because no one ever uses a 555 in a feedback loop, let alone in any kind of proper (e.g. peak current mode) SMPS control scheme. For which you'd reach for a UC3843 or better, but, who knows.
The "complication" is, CW mult. draws current on both phases of the waveform, whereas flyback proper only delivers current during the off phase. When the transistor turns on, it's hard* switching into the capacitive load of the whole mult. chain, so, a large peak current is drawn, particularly during startup. Which makes it awkward to use a multiplier with current-mode SMPS controllers/regulators. (The solution for those, is to use several secondaries in series, each individually rectified, to reduce the effect of winding capacitance. Not use a multiplier at all. Else, an LLC resonant topology is best, quite suitable to drive a multiplier.)
*But still not completely, as dI/dt is limited by transformer leakage, and peak current by that and total output capacitance. Which given they're using ceramic caps, the capacitance is probably pretty small, also meaning the output isn't capable of, heck maybe not even fractional mA, but surely not more than a few mA near whatever counts as "full voltage" output. (Which again, is complicated by the flyback phase being able to reach fairly high voltages, just at ever tinier currents; I'd consider "full output" to be something near a symmetrical switching waveform, i.e. peak flyback voltage maybe one or a few times the supply voltage.)
Tim