Gasoline equipment needs air. I think it is 14 plus grams of air per gram of gasoline. Depends on how you look at power density. Probably battery power density would be better compared to explosives than a fuel that needs air or oxygen.
I'm not sure what you're trying to argue. Yes, fuel requires oxygen....oxygen is abundant in our atmosphere so it's a non-issue. We're talking about energy density. How much work can be done by a specific volume of energy storage medium.
I did call it 'power' density when I meant energy density.
Batteries don't even come remotely close to the density of gasoline right now. What electric motors have is incredible efficiency though and instant torque at low RPM. So they do have an advantage along with less noise and less emmissions (depending of course on how the electricity stored in the batteries was produced). So battery powered tools have a very obvious benefit over gas powered tools assuming we can get the same or close to the same amount of work done without massive weight or wait penalties.