I opened the M5StickC and was kind of shocked of the obviously bad quality checks. 1. the 5V raiser pin to the hat connector wasn't soldered at all. 2. they are using a MP1541 switching regulator labeled as IB3GG to provide the 5V power. This is a heavy design fail and explains a) why the power never shuts off, but only drops to ~3,6V and b) why the batteries drain when shut off. Have a look at page 9 of the datasheet: https://www.mouser.de/datasheet/2/277/mp1541_r1.5-1398052.pdf
If the regulator is disabled power from the battery flows through L1 and D1 to the output. This is an explanation for another hardware revision using another switching regulator or placing a MOSFET before the IN pin triggert by the EN signal as well.