A weird question about the cutoff point of an RC low pass filter
Your confusion seems to come from thinking that the math suggests that there would be multiple possible outputs, but your intuition says that there should be only one.  The answer is that there is only one output, and here is why.

An RC low pass filter is a linear system.

For linear systems, the following is true.  

F(X1 + X2 + X3 ...) = Y1 + Y2 + Y3...

Which simply means that...

If input X1 produces output Y1.

And 

If input X2 produces output Y2.

Then

input (X1 + X2) produces output (Y1 + Y2)

A step response, which does contain all frequencies, simply produces an output which is the sum of the response of the filter at each frequency.  Note that even though there is an infinite number of elements being added together, their sum converges to a finite number at each point in time.  That sum is an exponential decay which eventually reaches the value of the step input.