Let it be known to others who encounter this issue:
To access animations that are imported with a hierarchy of meshes, you need to go to the dope sheet and cllick "add animation". This will generate a default controller behind the scenes and allow you to use the animation(s) the asset came with.
It is possible, however, that the object on which the Animator and Animation Controller are set are not creating the same effect you saw in the preview (since you have to select the correct part manually). If you find yourself frustrated trying to figure out where to put the damn things, you can re-key generic transformations yourself for objects that are parents of all the objects you want transformed (in my case, this was the Axel).
Like they say, if you want something done right, do it yourself (again).
Here is my first attempt at making gifs of my project. The seesaw and treasurebox have different animation periods and my hardware sucks, so there are noticeable jumps where frames skip or the animation doesn't loop seamlessly. Because I edited in Photoshop, the colors are further optimized and not representative of the actual game.