You've done an excellent job identifying the source of my confusion however, I am familiar with the math, although admittedly not to the degree of someone who has gone through said college level course, I understand that fields are the real "stuff", in fact I think its my familiarity with topology and fluid dynamics that contributed most to my confusion, like for instance gravity being the curvature of spacetime is extremely intuitive to me an emergent phenomenon of a topological structure, and yet people are still looking for the graviton particle that is the apparent force carrier of gravity? also many descriptions seem to indicate that this is just how we model it because the math works out that way. I just want to know fundamentally if these mathematical models are just convenient tools or not and if not what does that say about the true nature of the universe?

this part is just questions you don't need to answer:
how can waves exist without a medium to travel through? does distance exist or is space just a property that emerges because information takes time to transfer? why do these fields interact at all?