Why do things accelerate due to gravity at the same rate?

[ShayanJ]

Er, only that's not a reason is it? It's not a scientific explanation of any fact. It's nothing but a restatement of the fact (that "the acceleration of an object due to gravity doesn't depend on the mass of the object, so its the same for all objects.").

All it does is fix things right with the taught concepts and principles of mechanics? - (that I'm never sure don't contain some circularity or unobservables)

If you read OP's question, you'll see that it wasn't as obvious for him\her.

 

[pbuk]

What we have here is a failure to communicate - Josh S Thompson you do not express yourself clearly, and you don't listen to what people say. But to be fair, some of the people providing answers aren't listening to what YOU say either:

If I had a bowling ball and a ball the mass of the sun ...

In this case we clearly do need to take into account the acceleration of the Earth which. So to answer your original question (actually it wasn't a question, but let's ignore that for a moment,

The two balls would not accelerate at the same rate

From the point of view of an independent observer (more technically, an inertial frame of reference), the two balls would each accelerate towards the Earth at $\frac{Gm_E}{r^2}$ where $m_E$ is the mass of the Earth. However, just as the acceleration of the ball depends only on the mass of the Earth (and the distance between them), the acceleration of the Earth depends only on the mass of the ball. In the case of the bowling ball this is negligible, but for a ball the mass of the Sun it is much larger than that of the Earth - this is an important property of a celestial body known as its standard gravitational parameter: you can find some of these (including for the Earth and for the Sun) in this Wikipedia article.

So whereas if we drop a bowling ball from 100 ft we can ignore the motion of the Earth and calculate that the ball accelerates at about 9.8 ms^-1 and hits the surface in 2.5 s, if it were possible to place an object the size of a bowling ball but the mass of the Sun 100 ft above the surface of the Earth and release it, we can ignore the motion of the solar mass object and calculate that the Earth would accelerate at about 3,300,000 ms^-1 and collide in 0.0043 s.

So from the point of view of an observer on the surface of the Earth, the Sun would indeed accelerate 330,000 times faster than the bowling ball.

 

[pbuk]

It does the marble takes a longer slightly more arced (geodesic) path towards the Earth than the bowling ball.

That doesn't affect the downwards component of acceleration, or the time it takes to collide.

 

[TurtleMeister]

One of the reasons (and possibly the main reason) people have a hard time understanding the universality of free fall is that they confuse relative acceleration (acceleration relative to one of the accelerating bodies) with acceleration relative to the inertial frame of reference. This is what Nugatory was referring to in his post #22. The UFF is only valid when using an inertial frame of reference.