tl;dr: No, in fact bigger wheel may mean more effort
Answering a question similar to this allowed me to get an admission in my esteemed institute. I will try not to get too technical, here we go.
First of all (for sake of developing intuition), open the nearest window/door first with its handle, then close it, then open it again, this time use your palm keeping it as near to the hinge as possible.
You are very correct in saying that
C (circumference) is in fact 2 times
r (radius taken of wheel) and if we hide the constants we can safely say that
C is directly proportional to
In plain english, distance covered (irrespective of road conditions, be it urban, wavy, or even inclined road) varies directly as radius of wheel.
Now, think about it, you can cover 100s of miles with virtually no effort if you are sitting in a bus. You yourself realize going uphill will be difficult, why? 1 meter is 1 meter in all directions, clearly there's more to "work" than just distance covered.
So, when it comes to effort we must talk about force also and not just the displacement or the distance covered.
Now, work done is dot-product (think of it as regular multiplication with some added bells and whistles) of force and displacement. In our (slightly special) case the increased diameter/radius of the wheel implies that we need to exert more force to turn it (will edit n insert equations as soon as I learn MathJAX)
In plain english, big wheel implies more exertion to turn the wheel, even if we consider the wheel to be absolutely weightless.