You definitely need a school. If you solve your firewood problem, you’re very likely to have a tool shortage. An uneducated worker gets 1 iron per 1 surface ore, and 2 logs per tree. Then, the blacksmith takes 1 log and 1 iron to produce 1 iron tool. So for that amount of work you get 1 tool and 1 spare log. For educated, you get 2 iron units per surface ore, and 3 logs per tree. The blacksmith takes 1 log and 1 iron to produce 2 iron tools. So here, you get 4 tools and 1 spare log, for almost the same amount of work (with educated workers blacksmith gets materials for 4 tools, so they work twice). So you lose 75% of your tool production if you don’t build schools.

Firewood is a bit better. For uneducated you get 2 * 3 = 6 firewood per tree, for educated 3 * 4 = 12 firewood per tree, so you lose 50%.

After you build a school, make sure to always have at least 1 labourer no matter what – if your teacher dies and there isn’t a labourer to replace them, all the students will drop out. If you’re going for the Uneducated achievement (which I think is the hardest of all), you need to increase your production, and grow really slowly.