The HL is not used here to predict the peak production (i.e. when we gonna peak and at what rate?), Cantarell has already officially peaked. The HL is used here to estimate the URR and the logistic decline rate (K) (i.e. how fast we gonna lose Cantarell's production). The resulting logistic curve is simply modeling Cantarell's decline post-2005.
I understand this.  I am questioning whether the HL technique has any predictive value at all.  If it didn't predict Cantrell's peak, why would the technique suddenly become predictive immediately following the peak?

While I like the elegance and simplicity of the method, the shown fits do not convince me of its utility.

why would the technique suddenly become predictive immediately following the peak?
I see two main reasons:
  • we know that a fast and terminal decline is expected (see the official chart on Fig. 8).
  • the predicted URR (17 Gb) is close to the official number (20.5 Gb)

I could have used something else that a logistic curve: a gaussian or a constant R/P ratio used by economist. Any monotonic curve that is delimiting an area close to 9 Gb will have done the job.
Is there any reason to believe then that the HL method is better model than, for example, eyeballing the decline (based on the two post-peak points) and calculating the area under the graph to generate a URR figure?
You can find some background on the logistic model applied to the modeling of ressource depletion in the following post:

Links to tutorial material on Hubbert Linearization

It seems to work but no one knows why. I have a problem with understanding what the logistics K physically means. I think the real depletion rate for all Mexico is under 10% of available reserves using a more realistic rate model.

http://mobjectivist.blogspot.com/2006/03/mexico-oil-shock-model-part-2.html

Again, I don't think K makes any real physical sense.