This is a big question. The philosophical origins go back to the beginning of philosophical thought. The topic of spacetime has its origins in the theory of the aether. The topic of the aether and spacetime started to gain momentum with Galileo and the really picked up steam when Newton did his theorizing. It gained even greater strength in physics when Maxwell arrived on the scene and he used this physical model to develop his electromagnetic theory which was based on an aether model.
The aether hit its pinnacle with Lorentz’s quasi-elastic solid. Einstein developed the theory of Special Relativity based on the aether theory that he also believed in. Coincidentally, the Michelson Morley experiment was being conducted at the same time Einstein was developing his Special Theory of Relativity. Einstein was aware of Michelson’s Experiment data and knew before the release of Special Relativity that the aether wind was not going to be verified. Because of this knowledge he disavowed the aether when he released the Special Theory of Relativity which came out just a few months ahead of Michelson and Morley announcement of the now famous null result. The misinterpretation of the null result led everyone into believing since the aether wind was not detected that the aether did not exist. This led everyone in physics down into the rabbit hole. Where they have stayed now for over a century.
Later on Einstein changed his stance concerning the aether when he discovered the General Theory of Relativity. After 1916 he was again a believer in the aether and at this time the new name for the aether was coined. The new name was Spacetime.
Einstein attempted to pull everyone out of the rabbit hole in his speech dedicated to his mentor Hendrick Antoon Lorentz. The speech happened in 1920. Unfortunately it was too late as the new pioneers in physics were being led by the pied piper Neils Bohr (the champion of Quantum Mechanics) and they were now deeper than ever into the rabbit hole.
The last paragraph of Einstein’s speech.
Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.
I have analyzed this paragraph and realized that Einstein was the only one besides a few others like Lorentz and Fitzgerald that still had the correct model in hand. I describe this time period and the aether in detail in my book Master of Reality.
What I believe Einstein was saying is that “time” is an illusion (because time is an experience only and not a real thing). Time only exists in this physical realm because space itself is a physical thing. Space time intervals exist because space exists and has real physical metric (distance). Since space has a distance it takes a specific amount of time to traverse that distance. If space was not a real physical thing then it would take no time to travel through space. That is not what we see. So our experience has shown us that space is real because it takes a finite amount of time to travel to distant locations. The amount of time it takes depends upon what speed we are traveling. If space were equal to nothing it would take zero time to travel any where in the Universe. That is not what happens.
In my theory called Super Relativity the spacetime continuum is explored further and has been extended to the point where spacetime is something that can be modified by applying forces to it which then modify its properties such as tension and density these stresses in the medium affect the speed of light and the properties of permittivity and permeability. Because of this spacetime itself becomes relative.
So to summarize, the philosophical foundations of spacetime theories comes from classical physics and the great ones like Einstein, Maxwell, Lorentz and Newton. These men were the ones that were using the correct philosophical, metaphysical model. The correct model is the aether.