Go for it.
Then let's go step by step. Rather than post one step each time I will post a few and see where you disagree.
1 - First we define π as C/D.
2 - This can equivalently be written as C=2πr, where r is the radius of the circle.
3 - Now this is great for a complete circle, but what about an arc of a circle which only subtends some angle x?
Well, the circle is similar all around, so the arc length for x will be a fraction of the circumference, with that fraction dependent upon the angle.
4 - But what to use for the angle? Some people might immediately jump to degrees, but lets pretend we have never heard of them and need to make up our own.
We might be tempted to use an angle of 1 to indicate a complete revolution, but lets try a more natural approach.
We know that the arc length should be dependent upon x as:
l=k*x*C=k*x*2*π*r; where k is some constant of proportionality depending upon the angle, which is the inverse of a revolution. For example, if x was in degrees, then k would be 1/360 degrees.
5 - But what value should we use for k? Well, as we want to look at arc lengths, why don't we focus on the latter part of the equation and try to get rid of as many constants as possible?
That means we have l=k*x*2*π*r. What if we let k=1/2π? Then we have:
l=x*2*π*r/2π=x*r.
That sure looks a lot nicer.
6 - So we define our unit of angle as a revolution having an angle of 2π, or equivalently, that a circular arc of length 1, with a radius of 1 (or of length y with a radius of y) will subtend an angle of 1.
All this sound reasonable so far?
Poll