If you carry a few fire starters, you don't need bother with the axe and heartwood.
I carry "fire bunnies" (put a few in "da Box" last time it came around) which are nothing more than tightly wadded paper towels dipped in melted paraffin. They burn 5+ minutes and will light wet wood just fine.
I suppose if you are staying put for a week or more and dead fall starts disappearing, a saw and axe might be essential tools, but not for backpacking.