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To get blocks to render in-world, you just need to put the blockstate and model JSONs in the right places and name them the right things, and Forge will handle everything automatically. To get things (both blocks and items) to render in your hand and inventory, you have to put another model JSON in a different place AND put a couple of lines of code in your mod's base class.
That took me a while to figure out. But I did, so now all the things are working. Next step: Replace this derpily-textured cube with an actual 3D model.