No, I'd say the main advantage in getting fitter is that you can
expend more energy at a given task, although possibly there is an
increase in efficiency as well.

I must admit I do feel the benefit and find that even on my overweight
clonker I'm riding faster and able to push harder. In fact I'm at the point
now where I need to think about raising the gearing on the bike, it's granny
ring now showing signs of rust on it's wearing edges!

But actually probably the main advantages are a clearer head, more energy
for day-to-day tasks, improvement in my asthma (which was pretty bad),
increase in general strength etc etc... And I look rather less like Mr
Blobby than I used to!