Colpitts VCO by varying base voltage
"Can you identify which are the case and elaborate on why VCO designs are often so complex?"

Each oscillator has to fulfill Barkhausen´s oscillation condition (Loop gain slightly larger than unity for start-up and equal to unity for continuous oscillation). This condition must be met for one single frequency only!. For most of the oscillator topologies a change of frequency (tuning of the frequency dependent part) is connected with a change in amplitude - and, thus, alters the oscillation condition. 

There are only a few simple oscillator circuits that allow frequency tuning (over a relatively broad frequency band) without disturbing the oscillation condition.
Normally, as much as at least two active elements are required in those cases.  However, fine tuning often is possible with basic oscillator structures because corresponding (small) gain changes can be compensated by the non-linear amplitude stabilization (if existing!).      

EDIT: Some authors are using the term "orthogonal" to indicate that the described oscillator features independent control of the frequency and the oscillation condition.