Can you post a link to where you got the claims from. The comment re C1 does not make full sense.
JT (Joule thief) circuits are usually poorly designed or not really designed or show signs that the people who produced them did not have a good understanding of what they were doing. This circuit is in that class.
The LD1117 has a maximum input voltage of 15V. Higher than that will kill it.
LM1117 data sheet
The zener diode is meant to protect the regulator BUT it's voltage rating is lower than it should be.
The 1N4734A is a 5.6V 1 Watt zener. The zener voltage is too low to allow the LM1117 regulator to have adequate headroom at full current. It's likely that the "Joule thief" will not make enough power to let the LM1117 reach full rated output current.
The JT runs "openloop". If it makes more than 1 Watt it will try to destroy the zener and then the regulator and then the mcu. Without the zener, because the JT is a flyback converter the output voltage will ise until the energy available is dissipated. If the load is not accepting the available energy then the voltage keeps rising until the LM1117 starts to accept energy unintentionally (ie Vin_max exceeded).
Th C1 question's meaning is unclear. C1 may be fully charged without harm providing the voltages involved do not exceed the rated value os other connected components.
Overall this is not a good circuit. There are substantially better circuits available that do not depend on brute force dissipation of the converter output. Also, this circuit is not especially "designable" - it is difficult to say what the performance of the converter will be wrt power level or efficiency (but both are probably small).
My comment that better circuits were available was (very reasonably) queried in 2022 by @MicroservicesOnDDD
The OP's added second circuit is better as it provides feedback.
His 3rd circuit may be but is FET Vgson dependant.
A converter with more formal hysteresis turn on/off for the oscillator is liable to be more efficient.
If lowest cost or IC free designs are not a requirement then there are various ICs available that are able to be more efficient because they can address switching and regulation more properly.
An example only IC that addresses low power low-startup voltage applications is the LT3105. This is very expensive and while not ideal in performance, presumably represents state of the art when introduced. Datasheet dated 2019 - prior to that it was an Analog Devices product.
I don't know how the Joule Thief circuit's performance compares but it would be interesting to see as I'd class the 3105's performance as poor if it were not for the very low startup voltage.