Would an AC induction motor rotate if the power factor were unity?
Power factor is not "set", per se.  Non-unity power factor happens, in spite of the fact that we'd like it to be exactly one.

In the US, large installations like mills or factories are often charged both for real power and reactive power, at different rates (or they're imposed with a surcharge for non-unity reactive power -- it's the same thing, but shows up differently on the accounting sheets).  Such installations will go to some effort to bring the reactive power down, either with actual capacitor banks or with synchronous motors that aren't powering anything but are adjusted to look capacitive at 60Hz.