"The problem with this analytical method, however, is that it does not admit overlaps or co-extensions: one of the modes must prevail over the others, at different points in the text." This criticism is not justified: in the conception of Adema, myself and others, discourse modes/narrative modes are seen as prototype-categories with fuzzy boundaries. So actual text samples may well show features of two modes (i.e. show overlap). See also Allan 2013 in the Tsakmakis & Tamiolaki volume.
I do not see the contrast between a discourse mode analysis and a move analysis. Moves is a discourse unit consisting of one or more Acts, with a certain function in the larger sturcture of the discourse. One might see the discourse modes as a way of analysing the internal structure of a Move: which principle determines the coherence Move-internal Acts.