T2 upgrades are usually worth it, but more for durability. Pick damage increase is somewhat noticable but not really that "worth". Axes afaik have fixed chop damage so not only upgrade levels but also better materials dont really improve them much (especially since stamina cost increases, so bronze axe is best for almost all wood). They are a good sidearm so worth upgrading for that purpose, as well as durability.

Some low durability tools like cultivator i often upgrade to max, but otherwise if you are tight on resource, unupgraded tool is absolutely fine.