From the sounds of it, the doohickey was there on the NX-01 to cope with a new and exceptionally powerful warp field. One could speculate that the need for such a doohickey would re-emerge whenever an even newer and even more powerful warp field was introduced. That is, when the engines grew in power from TOS to TMP, the new field of the E-nil-refit (and E-B) once again required the doohickey - and when things plateaued again, the E-C was able to go the TOS route and abandon the doohickey.
The E-D might or might not have a field governor. It doesn't have blue domes, but it does have two clusters of blue squares atop the aft rim of the saucer. As discussed above, the presence of blue squares on starship hulls is sometimes associated with arboretums, but there's no canon proof for that - and blue glow more usually denotes the presence of powerful forcefields or subspace fields or the like. So perhaps those squares-of-squares on the E-D once again perform the function of the NX-01 dome? (They might also be the external manifestation of the saucer's warp engines, which "Encounter at Farpoint" basically insists are there, even if the TNG Tech Manual disagrees...)