That space is emergent and not axiomatic in matter time is a very tough thing to swallow I admit, but the math works with beauty and simplicity. For you (and everyone else on the planet less one) space is axiomatic and space is self evident and space seems therefore to be the most natural and intuitive of all of the axioms. Einstein certainly believed in space as an axiom.
However, the concept of space has its many problems dating back to Zeno's paradox and how space can be the infinitely divisible nothing that it seems to be.
"It doesn't have to be emergent because, lacking any physical features to explain, it doesn't need explanation."
Space has many features that need explanation and today we have quantum entanglement issues and, of course, gravity still resists the renormalization, which by the way, is an artifact of space. How do you explain the space inside of a black hole?
You and the whole world have simply grown accustomed to space as we all have, myself included. Space is a very deeply imbedded projection of our Cartesian mind and even though our projection of objects in space is necessary and very useful for nearly all of what we do...that is all space is...a projection.
The beauty and symmetry of zero and infinity, of a single point, a line with zero radius, an plane with zero thickness, and an empty volume are all very useful infinitely divisible idealizations, but that is what they are... Euclidean idealizations, albeit useful ones.
"Otherwise we are left trying to explain how space emerges from a dimensionless point, ie. the lack of space."
Space does not emerge from a dimensionless point...you see, that very statement already has the Euclidean logic of a point built in since there are an infinity of points that exist as an empty space. The action of a point moving becomes a line, a line sweeping is a plane, and a plane revolving is a volume. These are the actions that we actually sense as objects change or their image edges form details and then we project those changes as Cartesian objects in our mind.
In matter time, all objects exist as matter that is changing in time, i.e., matter, time, and quantum action predict all changes for objects without any direct use of space. You see, the universe itself is an object as a pulse of largely boson matter in time and we fermions are just along for the ride.
The three axioms of matter, time, and quantum action result in three dimensions as the norms of matter and time along with a phase that ties them together. It is from those three primal dimensions that the machine of our Cartesian mind projects the three dimensions of space.
So the question, "What is space filled with?" is preloaded with the answer-space must be full of something and the question already presumes that space exists as an axiom. The real question is what separates objects from each other and not what fills space. For the matter time scheme to work, though, the universe is an object that is made up of mostly boson matter, about 1e7 times more boson matter than fermion matter. Actually, the math works quite well...or at least I have not yet found a fatal flaw.
And it is such a pretty and simply theory of three, the trimal of matter, time, and action. One immediate consequence of this approach is that all action is Lorentz invariant by definition. All of the dilation effects of relativity apply to matter and time and so their projection into space means that space dilation therefore follows.
There are two things to look for in any TOE: falsifiability and utility. You need to have a way to falsify the model with an observation or test to show that it is more than just pure belief, and even if it passes that test, the model must still be more useful than what exists. Matter time passes both tests...