To explain, emergence as stated in the parent is a name for how a bunch of molecules could interact to make something you might call a football. This is a phenomenon at the "higher" everyday physical level emerging from the phenomena of the "lower" chemical level.
What I think this story misses is the importance of the "even higher" phenomena which resulted in a football being created, those of the human cultural level which caused the football to be manufactured and, for example, placed on some grass, or a desk, to be studied.
This isn't a purely sociological or philosophical point either, when you're looking at brain macro-activity for example, it's only partially explicable by referencing the micro-activity of, say, individual neurons, to get a better understanding you need to refer to activity outside of the brain that caused certain individual neurons to be acting like that. These higher level phenomena can be simple things like the movement in the neck that caused a change in stimulation, and also more wooly concepts like cultures which provide certain aesthetics or behaviours.
Emergence is important, but I feel its too often considered as a standalone phenomenon, rather than as one side of an irreducible causal cycle.
In my mind, the concepts of irreducible complexity and emergence are inherently linked through the concept of abstraction. An emergent phenomenon is a computationally dense phenomenon for which much of the computation can be abstracted away, at the cost of giving up a low-level understanding. In a sense, emergence is the antithesis of irreducible complexity. In another sense, emergence is a sparse chain of islands in a sea of irreducible complexity.
By "relativity" I'm assuming you're referring to SR or GR here. I always thought of relativity as an emergent phenomenon due to the large scales we are working with. What makes you say it is not?
We still haven't experimentally verified how gravity works on atomic scales, though, have we? I consider it completely plausible that it works differently than GR postulates, because to me (non-physicist here) GR seems to be a "large numbers" phenomenon and assumes space possesses certain properties that it may very well not possess on small scales (e.g. any kind of smoothness or even continuity).