We can orbit the view by holding down F4, or by holding down Shift+MMB (MMB = middle mouse button, or scroll wheel). Shift+MMB is usually more convenient; however, it doesn't provide a way to change the orbit center like F4 does. I propose that holding down Shift and doing a single click of MMB, without holding it down, should set the orbit center, point just like using F4 and clicking inside the reticle does.
That way we can hold down Shift, click MMB to set the center, then press and hold down MMB and perform the desired orbit about the correct center point.