why I want to transport wood

The Leave No Trace principles explicitly says "Do not build structures, [or] furniture". Yet that's exactly what I want to do as a bushcrafter.

One acceptable way to deal with this is to bring your own wood/roots/etc from outside preserve areas. (and then burn the wood or pack it out when you leave) Preserve areas are strictly for preservation, but there are places where it's more acceptable to harvest wood products, eg. fallen wood in urban areas, or private property with permission of owner.

However, you have to be careful when transporting wood products because of the Emerald Ash Borer quarantine.

regarding the EAB quarantine — can you move wood between states? between counties?

softwoods can be transported

However, note that softwoods are ok to transport. Only the ash trees are targeted for quarantine, however because ash is difficult to identify among other hardwoods, all hardwoods are subject to the quarantine. [1] [2]

It's not clear what they mean by "hardwood" though. There's a technical definition (presence or absence of pores) that can only be determined by careful sample preparation. And there's the layman definition that means "is it hard or soft?".

Well, the various ash species are all fairly hard. They aren't the hardest woods by any means, however they run from 45th to 70th percentile of hardness among all trees. So that might work.