You are right in the way you are thinking, but you don't have the full information: in a 600 kg plane the fuel is 100kg. The mass reduction does not make such a big impact, especially as you rarely top the tanks (you can carry more luggage or a heavier pilot & passenger) and you never empty it (30 minute reserve is usually raised to 1 hour reserve), so you have a 10% variation of the total mass of the plane on regular flights: no significant impact.

On the other side, large planes carry a lot more fuel as a percentage of the total mass, it really matters to them.

I hope to hear actual pilots chiming in, but from first principles:

It shouldn't be a large factor. Drag is independent of mass, and while lifting the fuel to cruising altitude takes energy, most of the flight should ideally be cruising.

The rocket equation doesn't apply; that's for an entirely different regime.

While drag is "independent" of mass, the lift required to maintain level flight is affected by weight, and drag is affected by lift. A heavier airplane will absolutely require more power to cruise.