I'm not sure that either of these methods count as traditional.

Air spades in particular are primarily used for rootwork, not transplanting. Bareroot methods are used for smaller trees. Bare rooting leaves roots in a very vulnerable state, so doing it on larger trees you intend to move and keep alive is a serious logistical challenge.

The most traditional method I can think of is "ball and burlap" where root balls are cut free in the field, and retrieved later in the season for final packaging.