There are a number of alternative theories to SR that posit absolute simultaneity (and hence reject SR's relative simultaneity). There are often referred to as Lorentzian relativity or neo-lorentzian relativity, and were originally developed by Lorentz and others (such as Poincare). They preceded SR by approximately 10 years.
These theories presuppose that there is only one frame (the "preferred frame" often called the ether or ether frame) in which the speed of light is truly isotropic.
In these theories, time is absolute and does not vary with speed. Here it is theorized that there is no actual "time dilation." There is only "clock retardation." Time, as an abstract concept, does not change. However the mechanical instruments which we use to measure time (clocks) are altered by motion.
According to such theories, the speed of light is variable, and differs from frame to frame (while time does not).
However, the speed of light is always MEASURED to be C in other frames if corrections are not made for the distortion of the measuring instruments caused by increased velocity.
The idea is that all (uncorrected) frames measure C to be the same because the rods and clocks used in those frames have been distorted by velocity. So, even though the MEASURED speed of light is C, the "actual" speed is not (except in the preferred frame).
These theories are often said make the same predictions as SR (although I would dispute that). Therefore, every experiment which is said to confirm SR can also be said to confirm LR.
Sexl and Mansuri did a several part, extended analysis which compared and contrasted SR with such alternative theories. Here is an excerpt from a wiki article which summarizes their work on the topic:
"So Mansouri and Sexl spoke about the "remarkable result that a theory maintaining absolute simultaneity is equivalent to special relativity." They also noticed the similarity between this test theory and Lorentz ether theory of Hendrik Lorentz, Joseph Larmor and Henri Poincaré."
The essence of your question was: "Could we work out a physics were time is absolute but maximum speed of light variable?"
The answer is "yes, we can (and have)."
Edit: I see now that I am basically just re-stating, in a different form, the answer given by Albert, above.
Edit: The second part of your question was "Why is the speed of light held constant here?." Because it is a (by definition unproven) postulate of SR. Einstein himself readily acknowledged that it was a "freely chosen" premise, not one that was dictated by experimental evidence.
One the other hand, Lorenztian theories posit the (also unproven) premise that the speed of light is not "truly" constant in every inertial frame, even though we measure it to be the same.