Actually, you don't have to include inertia for an intuitive explanation.
Imagine an astronaut up in outer space. He is working on the space station from the outside, floating about in free space. He then let's go of his wrench.
What would you intuitively expect the wrench to do? You would expect it to just hang there, right? Because there are no forces acting on it. Only when he grabs it again will it move, since he now applies a pulling force in it.
Our intuitive understanding is that things should not start moving unless acted upon by a force. This is one of the key ideas of Newton's first law.
Now hold on to this intuitive understanding and bring it into your bus scenario. A ball is lying still on the bus floor. Then the bus accelerates due to its engine forcing it forwards. But nothing is accelerating the ball yet.
So, the bus moves forwards underneath the ball and there's no reason to expect the ball to move along - not unless a force pulls in the ball, that is. And we do have a force pulling in the ball. That would be friction.
Friction pulls only in the bottom point, though, so only that point is accelerated forwards, not the whole ball. Because that point is connected to all other points in the ball, but slightly off of the centre-of-mass, pulling in this lowest point causes the ball to spin.
And there you have it. No force is pushing the ball backwards. The fact that it seems to move backwards is just an illusion - it is rather the bus which is moving forwards underneath it. And the forwards pull in the bottom-most point causes the ball to rotate (to roll).
That's it. There is no force "generated" that opposes acceleration of an object. The fact that the object resist acceleration is the expected intuitive default - that has then been named inertia and is represented (in linear cases) by what we call mass. Inertial forces or Euler forces which are typical names for the apparent "force" that pushes the ball backwards are merely illusions and are just invented to explain the seemingly backwards push from the non-inertial reference frame. Centrifugal forces are in this same category, just in the circular-motion case. From an inertial frame these forces do not exist - they are what is called pseudo-forces.
You mention frictionless cases and friction cases in the last sentence of the question. Note that for a frictionless case, the exact same explanation applies just without any rolling/spinning of the ball. The ball would simply remain where it is in space not being influenced by the fact that the bus is moving forwards underneath it.