Hi [NAME],   To my assumption (without any documented information, I can't be sure), the BIST button completes the circuitry of the SMPS (PSU) and it manages to supply power to the mainboard. I would recommend trying replacing the PSU first. I won't rule out mainboard as the root cause. 
The out of warranty 2018 PowerEdge T30 starts only when the BIST switch is kept pressed. Release the switch the machine turns off. Amber light blinks 2,2 in sequence.
PowerEdge T30, no power on