Even though the thin lens equation is messed up by the modern optical design tricks, the effective focal length can be calculated simply based on the focusing distance and the magnification factor of the image projected on the image plane, I guess?
The thin lens equation applies to complex lenses exactly the same as to simple lenses, in that for any complex lens there is an equivalent single thin lens. In a complex lens the lens equation is harder to use because you do not know where the nodal planes from which U and V are measured are located. A telephoto lens, like the 600mm, is a lens whose focal length is greater than its physical length in that
the rear nodal point is in front of the lens. The ordinary Nikon 70-300 extends its physical length by (roughly) 50mm in zooming from "70mm" to "300mm", so it is movement of the rear nodal point away from the lens that accounts for the extra (roughly) 180mm. It is not that zooming causes
the nodal point to move: zooming is
the movement of the nodal point.
As far as measuring the focal length by measuring magnification and focus distance, yes you can, but the problem is accuracy. If you use an optical bench, an accuracy of 1% - 6mm in the case of the "600mm" - would be a good result. If you don't use an optical bench the results will be very approximate.
From a photographic point of view, the important point is that a lens' focal length getting shorter as it focuses closer is not a design fault, it is a fundamental property of a lens. There are ways you can mitigate that effect - lengthening the lens, eg - but they have drawbacks.