Your metaphor of adding wet sticks to a fire is flawed, from a thermodynamics argument. Assuming that the fire is large enough, it will dry out the wet sticks through evaporation/boiling, and the wood's total contribution to the heat ("energy out") will be unchanged.
Now, of course for the people sitting around the fire, some of the heat that would be warming them up or used for cooking is instead used to force a phase change in the water, so it won't be as much "useful" heat, but that meaning of utility - which I believe is the one you want - is outside of thermodynamics.
For what it's worth, "for some very compelling anecdotal evidence" of the thermodynamic approach, see The Hacker's Diet at http://www.fourmilab.ch/hackdiet/ , which uses the "Rubber Bag" model. Unlike the garytaubes.com web site, the entire book and supporting materials are available for free download. The basic premise is the thermodynamic "if you eat more calories than you burn, you gain weight; if you eat fewer calories than you burn, you lose weight." (See also 'Food and fact' where "The rubber bag view of the body and considering only the calorie content of food is obviously oversimplified." and some of the complications are mentioned.)
I believe you are making a different argument, which is that you can't always measure the caloric impact of foods. Under the Hacker Diet, that makes little difference. It uses a feedback system based on weight trends rather than specific calorie counting. There's no need to know that oak combustion produces about 15MJ/kg and boiling water from room temperature takes about 3MJ/kg, because it only looks at 'is there enough heat'?