Technically that is sort of true, but its not a safe practice to take it literally. Combustion is a chemical reaction. It requires more than one elemnt to take place, if you start with O2 and end with O2 you have no reaction. The same could be said for Hydrogen and any other element. Oxygen compounds are flammable and that is what the danger is. thats the reason they put you in a tent at the hospital, its a controlled environment, so they know what is mixing with the oxygen.
The reason Oxygen doesn't burn coming right out of the tank is that the compounds it needs to mix with aren't usually (sometimes they are) present in the atmosphere. Acetylene on the othe hand, burns when mixed with oxygen, since there is already oxygen in the air, it will burn in natural air. So they call it flammable. Oxygen will do the same thing with many of the compounds present in a workshop, thats why you have to be really careful.
Oxygen doesn't really accelerate anything. It balances the equation. If you have 4hydrogen molecules and 1 oxygen molecule you end up with 2 unused hydrogen molecules. 4H2+O2=2H2+H2O+H2O. If you add another oxygen molecule now you've got enough for a complete reaction. 4H2+2O2=H2O+H2O+H2O+H2O. The complete reaction is much stronger!
Thats what your doing when you tune the flame of a torch, balance the oxygen and acetylene for a complete reaction.
So while its true that if you started with a vacuum and pumped it full of pure oxygen, it wouldn't burn, its not safe to say that oxygen won't burn and go pumping your shop full of it! You don't know what it will mix with