Can I make a VCO full audio range with only one capacitor?
Yes, you can get 20-20000 Hz with linear control voltage output from something very similar to this circuit with a single capacitor. 

Substitute a CMOS-input RRIO op-amp with a low Vos guaranteed to work at 3V (and with GBW of at least 5-10MHz) and substitute a MOSFET such as the DMN5L06K for the BJT + 10K base resistor. 

If 3V = 20,000Hz, then 3mV  = 20Hz, so keep the maximum offset spec of the op-amp at maybe a few hundred uV. Microchip probably has an inexpensive part that will suit. 

It's a bit easier with a higher supply voltage such as 15V because the offset requirements are less stringent, so the BJT and op-amp offsets don't matter as much.  

By the way, rather than coming up with V+/2, simply split the right-hand 51K resistor into two 100K resistors as a voltage divider between V+ and ground.