Philosophically I don't know if this question has ever been answered satisfactorily. But over the last 100 years fields have become so mathematically convenient that they're too hard not to use.
The problem Einstein and others grappled with was different, and concerned the fact that according to relativity nothing should exceed c. Entangled particles seemed to violate that principle, because the collapse to a given pair of states was instantaneous. The modern resolution to that is more practical than philosophical, and points out that information isn’t transmitted faster than light in entangled particles. Information propagates at or below c, even though at some scale entangled particles are statistically correlated. Relativity is only violated, and causality threatened if information is transmitted FTL, but entanglement doesn’t allow for that. You need a classical channel make entanglement work for communication, and that classical channel is going to be no faster than light. Without a classical channel, entanglement just gives you random noise.
But none of that answers the 18th-19th century Physicists question about how fields extend across the universe, or how light propagates in empty space. What the hell is a field anyway?
Also, it doesn't explain how the expansion of space happens, what made inflation happen, and what dark energy is.
Now, it is very easy to get hung up on this "Is it real or not" thing, and most people do for at least a while, but please just put it aside. When you peer really hard into the depth of the theory, it turns out that it is hard to say for sure that stuff is "stuff". It is tempting to suggest that having a non-zero value of mass defines "stuffness", but then how do you deal with the photo-electric effect (which makes a pretty good argument that light comes in packets that have enough "stuffness" to bounce electrons around)? All the properties that you associate with stuff are actually explainable in terms of electro-magnetic fields and mass (which in GR is described by a component of a tensor field!). And round and round we go.
Light doesn’t propagate in empty space, because space isn’t empty, it’s permeated with quantum fields in constant flux. What light doesn’t require is a medium, and a field is not a medium. The problem here is that these theories are really described in terms of a bunch of math, and when we use words to describe those theories, something is lost in translation. There are no words that fully capture what a field is in QFT, you must use math.
Once you adopt that perspective, inflation would just be the result of another field with particular properties (esp. a given range of potential energy curves). Of course, inflation is an untested theory, and there are competing ones which try to explain the apparent homogeneity and isotropy of the universe. Inflation is probably the most popular, but it doesn’t rest on the kind of firm foundation that QED does.