The LRC circuit (which opens automatically in Falstad's) is a mere ringing circuit. You admit a burst of current, which causes it to swing back and forth a few times. However it has no amplification or feedback, therefore it does not sustain oscillations.
The Colpitts is an oscillator because it has feedback and an amplifying stage. The action is adjusted so that it gives a kick during each cycle in the LC (or LCC) tank circuit.
I have collected a few oscillator circuits built around a single LC tank. They are often difficult to get going and to keep going. I believe the reason is because the LC tank does not provide a voltage swing which is easy to use at the precise moment, when it could turn on a transistor and admit the necessary small burst of current at the moment it is needed.
The Colpitts has the split capacitor, which provides a node where the voltage swing rises at the precise moment, when it turns on a transistor at the precise moment, etc.