I have troubles to understand how lower bounds w.r.t. circuit complexity and upper bounds w.r.t. uniform machine models can be used to show completeness results.
For example, the word problem for regular languages is known to be $NC^1$-complete under $AC^0$-reductions and in ALogTime, which equals DLogTime-uniform $NC^1$. But the latter is not known to equal $NC^1$. Maybe I am misunderstanding something about the $AC^0$-reduction, but I would not think that it is powerful enough to "handle" the non-uniformity? This would be the only explanation I can come up with.