One formula involving the gamma function is the Gauss multiplication formula:
Note that this formula is simpler with tau than pi.
The special case of this formula when n=2 is Γ(z)Γ(z+1/2)=sqrt(τ)*2^(1/2-2z)*Γ(2z)
Now, by setting z=1/2, we get Γ(1/2)Γ(1)=sqrt(τ)*2^(-1/2)*Γ(1)
Since 0!=1, Γ(1)=1, and we simplify the equation to get
So the only reason that Γ(1/2) is the square root of pi is because we used a special case of a more general formula which is simpler with tau, but in this special case, a factor of two happens to cancel out with the 2 in 2π, so the pi really is half-tau there. The constant that simplifies the general formula is more natural, not the constant that simplifies one case of it due to factors of 2 cancelling out.