Everywhere I look for these stones I can only find crackpot conspiracy theorists that think it’s an ancient spaceship. Is there any real reading I can do on this subject?
I suppose you mean Ishibutai Megalithic tumulus ('Ishibutai Kofun'), right?
Main archaeological excavations of this monument was conducted by the project team of Kyoto University during 1930s and 1950s, and probably due to the fact that they were reasonably extensive in accordance with this period's standard, little additional excavation has been done after these two main ones.
A part of these excavations report (pictures and films) can be checked at: http://exhibit.rra.museum.kyoto-u.ac.jp/ishibutai/index.html (unfortunately mostly in Japanese only)
As for the historical background of this tumulus, we know little to be added since these two excavations. All we can say is:
(Added):
I afraid we have difficulty in finding any reliable English introduction to Kofun Period Japan, so, as a general introduction (unfortunately not so useful for Nara-Asuka context in OP, though), I recommend the following English translated interview with the professor specialized in Ancient Japan: https://www.meiji.ac.jp/cip/english/research/opinion/Toru_Wakasa.html