On the topic of the classes and the keywords attached to their abilities, my understanding of it is that there are no hard restrictions on what sort of keyworded abilities a class can equip (so, a tempest can equip wizard abilities, even though none of the wizard abilities are Nature, Physical, or Lightning tagged). However, gear and passives largely work by modifying abilities containing specific keywords (so, "all Arcane abilities get buff x," "Fire abilities are modified to do effect y;" stuff like that).
So, the tempest that only uses his own class's abilities, or abilities from "related" classes, only needs to worry about gearing to optimize for the three keywords of Nature, Physical, and Lightning, while one who multiclasses into a wildly different class starts adding to the list of keywords they need to worry about gearing for (of course, there was also that example item they showed that made Lighting skills also count as Physical, and those types of items open the door for the more off-the-wall builds, but I imagine most people will start by keeping their builds "in-theme" for their base class, to make it easier).