The transistor needs a DC voltage between its collector and emitter or drain and source in order to work. What will happen when you put a capacitor in series with the resistor to the collector/drain is the capacitor will initially have 0v across it, but it will quickly charge to the supply voltage through the transistor, leaving 0v across the transistor.
For example, this circuit
produces the following output:
The charging of the capacitor and subsequent drop in collector voltage is visible on \$V(cap)\$, and the waveform for $\V(out)\$ is quite distorted due to the capacitor.
However, if I replace the capacitor with a short circuit, a mostly distortion free sine wave is visible on \$V(out)\$, with an amplitude of about 2V pk/pk.
Finally, for small signal amplifiers using BJTs, the transconductance \$g_m\$ (and input resistance \$r_e\$) is dependent on how much quiescent current is flowing through the transistor.