(My guess).
The idea could possibly be that wisdom-less authority simply does more damage than work for its intended purpose. In part, authority is meant to govern and, therefore, polish a system to ensure the consistency of its effectiveness in the future.

Now, the edge-less axe reference probably speaks to the same idea, being that of a tool which is malformed and therefore inadequate for the job--more fit to bruise. And the reference to "polishing" is more so speaking to the role of authority, not an axe's ability to polish anything (Something tells me that's not exactly what Bradstreet was getting at). Instead, this part is probably talking about how it is an authority's job to "polish" and that a lack of wisdom can dull its edge, or effectiveness.

It doesn't seem like she's saying that axes polish anything but is actually saying that an authority is meant to. Only when that authority is wisdom-less will it recklessly bruise a system/society/tree.

I suppose that the "bruise" example speaks exclusively to the function of an axe without an edge, while the "polish" example speaks exclusively to the function of an authority figure. And for me, this seems like the only way I can understand the context of this quote...I hope it gives somebody some better ideas.