Why does this amplifier not work past a given Vcc?
The problem with your circuit is the 2N3906 transistors are wired backwards. In this configuration they have very low gain, but the real problem is that reverse bias between the Base and Emitter exceeds the breakdown voltage, which is not modeled in the simulator. 

According to its datasheet the 2N3906 has a minimum Emitter−Base breakdown voltage of 5V, which must not be exceeded. In practice it acts like a Zener diode with a typical breakdown voltage of ~7.5V. In your circuit this clamps the voltage at the lower end of R1/R10 to ~7.5V, loading down the driver stage and reducing output voltage. Using a lower value Base resistor makes it worse because it pulls more current through R2/R5 which lowers the drive voltage even more. 

Fixing the problem is simple, just swap the Collector and Emitter of Q3/Q4. For the future, remember that simulation is only as good as the models used, and most models do not simulate accurately when the device is operated outside its specifications.