Power Supply Current is very low
Holy cow, talk about an undersized heat sink.

Your 24 VAC transformer will produce about 32 volts at the main filter caps. Producing a 5 volt output from the LM317 means that you are dropping about 27 volts across it. A current of .14 amps says the LM317 is dissipating  about 3.8 watts. The data sheet for your heatsink says that the temperature rise AT THE HEATSINK will be about 80 C, for a heatsink temperature of ~105 C. This is consistent with your measurement. If you look up the datasheet for the LM317, it specifies a thermal resistance (chip to case) of about 5 degrees per watt, so 4 watts on the chip will give it a 20 C rise over the case. Since the case is tied to the heatsink, your chip temperature is right about 125 C, which is the absolute maximum allowed. Dave Tweed is right about the regulator shutting down in order to keep from self-destructing.

If you could pull 1 amp, you would be dissipating 25 watts, and your heatsink temperature would be (roughly) 500 C. Plus another 125 C rise to the chip itself, and there is simply no way that's going to happen.