Corrections The voltage swings between +12V and 0
Good. Because a negative swich WRT GND is impossible. The H-bridge can only switch between 0 and +12v. They way it reverses the motor is to switch which pin it is sending the 12v to.
Measuring between two terminals of M1, M2, M3 and M4, the voltage is zero
That is not a good sign.
When I measure either one of the terminals of M1 W.R.T. GND, the voltage swings between +12V and -12V.
That is curious. If the voltages on the terminals are ramping up and down, but the voltage between the two terminals is 0, then the two terminals must be ramping up and down together. But based on the code, they should be 180 degrees out of phase. I can understand how half of an H-bridge might be damaged and not respond. I'm not sure I understand how it would do what these measurements indicate.
There is no voltage measured at all for M3 and M4 using any measurement points.
I guess we can assume that the second L293D chip is dead.
I believe you mentioned that you had two of these boards. How does the other one perform?