Making a low voltage oscillator audible
Do you see 2.5VDC across each of the 4.7K resistors on your divider?  If not, something's broken.  (Verify that your battery still puts out some volts even when connected to your circuit.  Does it drop far lower than 5V when connected?  Dead batteries tend to do that.)

Note that it's not a piezo speaker, it's a piezo beeper, and designed not for the usual 2KHz of typical beepers, but for the highest available 4KHz resonance.  

If it's running at 477, you shouldn't hear anything down there, since it's off-resonance by 8X and down by 20dB.  (You might hear the edges of a square wave: some high, very weak harmonic.)

Change your R1 C1 to give far higher frequency output than you now see.  Maybe a 0.01uF cap, and a 10K pot for R1, or just keep the 0.1uF and set R1 to a much lower value.  Tune for max loudness from the piezo transducer.  The square-wave output should then give louder results than the triangle.  And, note that R3 is not for frequency control, since it greatly reduces the triangle peak as it increases frequency.  Better to leave R3 value slightly below R2 (to give switching and oscillation.)  Then use a pot for R1.