Good point Tony. I remember a Single Board Computer I used in the 80's that worked with a switching 5V supply, but would not work with a linear one. I was controlling motors and they ran away. I was able to fix it by replacing an inverter IC with a schmidt trigger version. The IC was socketed.
RESET sometimes needs a good edge (fast rising) trigger.
Simple reset circuits might just be a resistor to Vcc and a capacitor to ground. They are unreliable.
I'd try replacing that with a real POR (Power On Reset) IC or a Voltage Detector.