If you accept that what we are most interested in, is reduction of temperature, then it makes sense to define cost in terms of temperature.
When we do that it becomes clearer why the energy cost isn't what is important, it is the temperature cost which is important. In the scenario that we are using energy which is free per Joule, and use of that energy has negative temperature cost, encouraging us to use not less of the beneficial energy, but more if it, it makes no sense to introduce anything which results in temperature cost overheads per Joule of energy storage capacity, if we don't have to.
Otherwise we would be removing any benefits of having energy which is not only for free per Joule, but also free per Joule capacity storage.
In the case of hydrogen there is no direct relationship between energy storage capacity and cost. The amount we can store in hydrogen has no such limits. There is a relationship, but it is influencable by development of storage technologies - tank construction improvements for example.
Another way of seeing this, is by saying temperature cost, is effectively energy liberated, like CO2 was liberated by fossil fuels energy use.
The more energy we store, the more we bring back down the temperature by capturing energy back into non heat form.
This is a little like carbon capture. What we are doing is energy capture.
We are replacing the enetgy capture function which was already done by nature in the form of fossil fuels. Our human equivalent of that is hydrogen storage.
In the positive energy use scenario, then you can see that the object should be to encourage storing as much energy as possible.
With batteries, we can't get around the fact that there is a fixed relationship between extracted material mass, and Joules energy storage capacity, depending on battery chemistry.
This is also why batteries as stand can't compete with hydrogen in the case of usage for aerospace.
The storage capacity weight cost is what is important there, and it too has overheads of mass per Joules storage capacity we can't get around.
Hence why the solution is to use hydrogen, rather than batteries, though there is still requirement to use batteries in some applications, we should work towards minimising or eliminating their use where possible.
Does this make sense to you? I don't know of Barnard analysis but it sounds like it might miss the point. Is he a Systems Engineer?