He bought another on-off switch and I swapped them out. There were two
white wires that I connected to the brass screws and two red wires that
connected to the silver colored screws on the sides of the new switch...
just like the old switch was set up. Initially, it seemed to work just
fine but we later found it now acted as a sort of master switch. The
other switches were no longer able to turn on the lights unless this new
switch was also turned on. What in the hell happened?

Did you purchase a 4-way switch?