I do wonder what that would look like—I think it makes sense for him to have Lightforged armor and to potentially even integrate it into the higher levels of the Alliance, but I suppose he could use a more Alliance-traditional set of armor.
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I generally agree—I think that Turalyon seems harder than Anduin, but he's not a frothing fanatic. He's always seemed fairly tame and could put aside his frustrations with all sorts of things—he seems to still care for Alleria despite her change in course, he tolerates Orcs when necessary, etc. He is a fairly pragmatic and tame person, all things considered.
I think his change in the course of the Alliance's leadership would probably simply be towards a more militant approach, but not strictly a lunatic dictatorship Garrosh-style. I think he'll be more assertive and try to remobilize them, potentially even restoring their original military hierarchy and move away from the royalist approach they currently have. He was, and is, Supreme Commander, and I figure that the "High King" thing may not even last for very long. I do expect a more assertive, less bland-as-paint Alliance that actually does anything, but I don't think that they will go full Garrosh. If anything, the Alliance will simply become better from a narrative standpoint instead of Anduin and Friends going around making everybody get along.