Voltage source connected to a MOSFET source
I don't think there's anything incorrect about the first circuit - sure it's trickier to get a stable bias point but as you've not shown where the bias comes from I could make the assumption it is lightly tied to the drain - this produces negative feedback (as well as bias) and stabilizes the circuit.

For the 2nd circuit, without the capacitor, the voltage source will steal all the bias current and turn it into exactly the same as the first circuit.