I should have been more precise: it is the velocity of the crank that matters, not the velocity of the bike. The cadence is, up to a factor of 2 pi, the angular velocity. And since you know where on the crank that sensor is, this gives you the linear velocity of the crank. From that you can compute the power as the (dot) product of force and (linear) velocity.

I was only referring to the strain gauge when it came to temperature compensation. Elite’s optical sensor is, as a matter of principle, not affected by temperature fluctuations, so it doesn’t need to compensate for that either.

No, it is not an estimation of power, you compute power from the quantities you measure, just that you are using a different equation for power. Crank-based power meters use power = force * velocity, where * is the dot product of vectors, and is based on the mechanics of the crank. But all ways are equally valid. Other power meters use an energy balance, which means they measure different physical quantities. In case you have a rotating flywheel, once you account for the friction built into the mechanism and all other interactions, you can make an energy balance. From that you can infer the contribution by the rider in the way that @LucasKun explained.

Neither do power meters. Strain gauges infer a force from a change in resistance, which, in turn, is due to a change in length as a result of the applied force. Crank-based power meters also need cadence as an input, and from that they compute the power. But it is important to keep in mind that strain gauges do not measure power.