Current source will consume headroom as well!
The intention of using a resistor here is not to have huge impedance as that of a current source.
The resistor is used only not to let the current go haywire. For the high purity VCO's that you are speaking about, the currents would be high and limit the resistor to a low value. It is a compromise., if you are OK with the power consumption without the tail resistor, that would be great, but otherwise it is a simple solution.
Try to use a resistor with CTAT characteristics so that the current increases as gm decreases due to temperature increase.