Edit, mostly deleted what I wrote: I wrote a bunch of nonsense here because I focused on the possibility of D_z potentially containing some nonzero elements, when the point was that it should presumably be a zero block.
Yes. I do think this works if E_{zz} is nonsingular. But I do not believe that can be guaranteed. So this is not fully general.
The Fix-Heiberger reduction starts in this way but then makes a further rank decision on E_{zz} and further refines the partition of E and D to deal with the possibility that E_{zz} is singular. It probably would work somewhat robustly in this case, but it’s a more complicated algorithm and I don’t think it ever made it into LAPACK.