Truthfully it would not make a huge difference. But, if you look at some of your formulas.
For sine & cosine. π radians for a half cycle, 2π for a full cycle.
Circumference of a circle: πD = 2πr
Area of a circle: πr^{2}
Volume of a sphere: (4/3)πr^{3}
Say you introduced a new value, Π=2π
Then your radians might be simpler, with the new Π radians for a full cycle, although you still have the half cycles, and etc.
Your circumference of a circle would be a wash as you would ignore the diameter, and just calculate everything with the radius.
Your area of a circle suddenly has an additional step: Area would be (Π/2)r^{2}
And, your volume of a sphere would be a wash too... changing from (4/3)πr^{3} to (2/3)Πr^{3}
I presume the reason for the choice of π rather than 2π in the past was that multiplication operations were easier than division operations. So they chose to minimize the division operations by using the current π.
Do math textbooks still have interpolation tables? [xx(]
Now, everyone either solves equations symbolically, or punches them into a calculator, and thus the multiplication/division distinction is far less important.
Anyway... since I like Apple pi, Blackberry pi, Rhubarb pi... well.... I'm just a big pi fan. Why change a good thing?