The reason that the acceleration is independent of the friction for an object rolling down a plane (assuming no slipping) is because the friction in this system is static friction, and it does no work on a rolling object. Consider the following diagram:
The point of contact with the surface denoted by the red dot only comes into contact with a single point on the surface while rolling. Since friction only acts when it is in contact with the surface, and the distance the point travels while in contact with the surface is $\Delta x = 0$, the friction acting on the object can be considered to be static. The static frictional force is generally written as $F_s = \mu_s N$, where $\mu_s$ is the static frictional coefficient, and $N$ is the normal force. This isn't strictly true, and should be more generally written as $F_s \leq \mu_s N$. So, the magnitude of the force can change depending on the incline to a value necessary to supply the needed torque to get the object rolling.
Now, lets consider the work done by a static frictional force. The work done on a system can be considered to be how much you change the energy of the system. Since static friction does not change the energy, the conversion of gravitational potential energy to rotational and translational kinetic energy remains the same regardless of the coefficient of friction, $\mu_s$. Therefore, the acceleration is not affected by it. So, how do we know it does no work. Consider the following definition for work.
$W = F\Delta x$
where $F$ in this case is the force of friction, and $\Delta x$ is the distance over which the friction acts. Now, you might say that since the object is rolling, the frictional force acts over distance the object rolls, but this is not the case as we've established, because the point of contact with the surface for the rolling object does not actually move relative to the surface when it is in contact with the surface.