The 120 volts your digital meter sees is phantom voltage, not really able to power anything.
That's why I got one of these. It's self-powered and pulls about 10mA at 120V. Enough to trip a GFCI, enough to be sure that it isn't phantom voltage.
https://www.homedepot.com/p/Klein-Tools-Electronic-AC-DC-Voltage-Tester-ET60/301107510