(1) The net work on an object that rolls without slipping can be exactly divided into a "work on the center of mass" and a "work causing rotation about the center of mass": $W_\text{net} = W_\text{com} + W_\text{rot}$.
In other words, for a macroscopic object (which should be thought of as rigid body composed of $N$ connected particles) the net work on that object is well-defined as the sum of the net works on each particle, and that sum can be decomposed into two such-described parts:
$$
\begin{align}
W_\text{net} &= W_\text{com} + W_\text{rot} \\
\sum_{i=1}^N W_{F_\text{net,$i$}} &= \int_{t_i}^{t_f} \vec{F}_\text{net,ext} \cdot \vec{V} \, dt + \int_{t_i}^{t_f} \tau_\text{net,$z$} \, \omega_z \, dt
\end{align}
$$
where $\vec{F}_\text{net,ext}$ is the sum of the external forces on all particles, $\vec{V}$ is the center-of-mass velocity, $\tau_\text{net,$z$}$ is the net torque on the object about the axis through its center of mass, and $\omega_z$ is the angular velocity of the object about its center of mass. This assumes a circular cross-section, such that the rotational axis passes through the center of mass.
I have proven this at the end of my answer to the above-linked question. The question was essentially about a claim by Kleppner and Kolenkow that work should be treated this way for an object rolling down an incline (although KK did not prove it to be true). The statement is claimed in some form in many textbooks, but rarely is it applied explicitly, and I have not seen it proven, though it it quite simple to do. (Perhaps it is too advanced for an introductory textbook but too basic for a Classical Mechanics textbook.)
I should also note that a work-KE theorem, with distinct center-of-mass and rotational parts, is an immediate consequence of the above equation:
$$
W_\text{net} = \Delta K_\text{com, trans} + \Delta K_\text{rot}
$$
where $K_\text{com, trans} = \frac{1}{2} M V^2$, with $M = \sum m_i$, and $K_\text{rot} = \frac{1}{2} I \omega^2$, with $I = \sum_i M_i R_i^2$ and $R_i$ the distance to the axis of $m_i$.
(2) Although it is true that the work done by static friction is zero on an object that rolls without slipping down an incline, it is not because "the point at which friction is applied does not move."
Using the above expression for the net work on the rolling object, one finds that:
$$
W_\text{net} = W_\text{com} + W_\text{rot} = \left[\left(F_g \sin \theta - F_{fs}\right) \ell \right] + \left[F_{fs} \ell \right] = F_g \,\ell \sin \theta
$$
where $\ell$ is the distance traveled and $\theta$ is the angle of the ramp. Therefore the work done by friction is zero because its contribution to decelerating the center-of-mass exactly cancels its contribution to accelerating the rotation about the center of mass. This was the result shown by Kleppner and Kolenkow and, interestingly, highlighted in a paper published last month in Physics Education.
The claim that "the point of contact does not move with respect to the surface, and therefore $F_{fs}$ does no work" is perhaps the most widely repeated claim about rolling friction (I remember teaching it just a few months ago), but it seems, to me, to be simply false. The "point of contact" is, I think, a red herring. The net work on the object is, as written above in statement (1), the sum of the net works on all particles. So, one could say that the particle in contact with the surface is not moving at that moment and therefore there is no work done on it. But for the rigid body, the effect of static friction carries through to all other particles through their internal forces on each other. These internal forces explicitly cancel out, via Newton III, in the calculation of $W_\text{com}$, but they remain in the calculation of $W_\text{rot}$.
(3) When a car accelerates on a flat road, its kinetic energy is changed by the work done by static friction on the rolling tires.
I don't see how this can be a controversial statement. The only horizontal external force on a car (neglecting air resistance, and assuming the tires roll without slipping) is static friction. Therefore, when it speeds up or slows down it is because of the static friction force. But, if one accepts that "static friction can do no work because the point of contact is not in motion with respect to the ground", then I don't understand how one can explain the change in kinetic energy of a car when it accelerates.
