They don't take over "whole dimensions", if you mean "every last object in them". Honestly, we don't even know that the TSAB has "conventional" interplanetary space travel (that is, not traveling between dimensions). On the other hand, if hundreds, thousands, or an infinite number of planets may be reached by cruising for a little while through dimensional space, why bother with the long route?
Does Nanoha have a Messiah complex? Maybe a little. By adulthood, she really does have the idea that she and her close friends have the power to "save" the world. On the other hand, that's "save" it from actual threats/invasion/things they can make go boom, not "from itself" or anything metaphysical like that.
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