Barbers are meant to be a thought experiment mostly: even if there were in fact no rise in productivity their wages would still have gone up in line with the general rise in wages. So productivity rises in one part of the economy [less than or equal to the set of non-barbers] spreads out.
Now let us imagine that world production is energy-constrained. [In fact it is only oil-constrained, but that is a complicated story since oil is substitutable, with difficulty, by other energy. Note that the substitution so far has nearly all been coal, with the rise in renewables unable to noticeably slow the rise in coal use, let alone reduce the total.] In an energy constrained world it is important to look at the productivity of energy. Let us first assume that the productivity of energy stays fixed. Now suppose there is a small rise in worker productivity in some section of the workforce. That means those workers get more "right to consume" than they had before [money is a "right to consume" token]. But by assumption that extra stuff they consume uses more energy, and there is only a fixed amount of energy with a fixed productivity. So the rest of the workforce has to get less consumption. So as far as the know-nothing economists sees it: all other workers have lower productivity, and total worker productivity doesn't change.
Now look at it the other way. Say there is a rise in the productivity of energy (i.e. a rise in energy efficiency). A little thought shows that total production goes up. However there is no need for this to appear as an increase in worker productivity if, as at present, there are lots of unemployed. Instead we can increase production at the same worker productivity by expanding the workforce.
In summary: we are used to a world where worker productivity increase requires more energy to be effective, and that extra energy has become available. This leaves us ill-equipped to think about economics in a world where that doesn't happen.