While I don't disagree with what you have said, the question of stability - conditional or not - would seem to be somewhat outside the OP's understanding - and well beyond the question asked.
The fact is this: Inductors and capacitors can in some cases be used to achieve similar goals, but operate on completely different principles and, therefore, are not interchangeable in any way.
Indeed, the space within which exchange is possible, is rather abstract compared to simply swapping out one component for another.
Namely, if you swap:
Voltage source <--> Current source
Series branch <--> Parallel nodes
Capacitor <--> Induuctor (value = 1/x)
Resistance <--> Conductance (G = 1/R)
You'll end up with the same RLC network, but with a series-parallel transformed equivalent.
Note this is not generally possible with nonlinear circuits, because we don't have every possible combination of device: a diode has a constant-voltage characteristic in forward bias (transform: current regulating diode -- a boutique item, though something which does exist); a transistor has a voltage sense input to current source output (a transconductance amplifier), where the input and output exhibit parallel capacitance.
Tim