No need to pay for research costs, and for one farming party you need minimum 5 swords or 3 axes. 3 axes cost 180 wood, 5 swords cost 150 wood. More than balances out the extra time each run, imo.
Assuming you send out groups of 20 spears, after 20 hauls with sword protection you would have time to run 24.5 axe runs, which which creates a surplus that covers axe the additionally cost of research.
Axe are much more effective add handling spikes than swords. Play around in simulator...the effect is huge; often axes will cut losses in half.
Also you ignored how much iron swords consume, iron that will be needed for lc shortly.