..... by running a wire from the battery to one terminal on the brakelight switch, and applied petal. Hey!! they work. The turn signals work the way they should, so can the problem still be the switch?. .
Were the turn signals not working either?
With a power jumper wire to the brake light switch, if both R & L brake lights turn on and off with the pedal the brake light switch is fine (and also the brake light connections thru the turn signal switch).
Trace the original power wire for the brake light switch back to it's source, checking for power and loose connections. Power to the brake light switch may come thru a circuit breaker in, and a battery '+' connection on, the headlight switch. (my wiring diagram book is for '55/'57)
(edit/followup)
Found a '61 Galaxie wiring diagram here...
http://www.oldcarmanualproject.com/tOCMP/wiring/5765wiring%20diagrams/Ford/MWire5765-206.jpg
Power to the brake light switch comes thru a green/yellow wire from the headlight switch. Also under the dash is a junction/connector in that wire that sends power thru the door switches to the dome light.
Turn signal power comes thru an orange/yellow wire, also from the headlight switch, to the flasher.
The headlight switch has internal thermal circuit breakers for the 4(+) lighting circuits that go thru it, each indicated by the
|~| symbol within the diagram of the switch.
.