Rodin. Let's do Special Relativity first.
1) You are correct. It is possible to define a "Universal Time" (UT), and with such time,
the speed of light is not constant.
2) However, if you define and then use this "Universal Time", the electromagnetic equations
become nasty when transforming between UT frames and they "blow up". You do not have well-behaved
force equations and UT EM fields in the new "UT coordinate system" are really unusable!
3) By contrast, if one uses "light-travel-distance-divided-by-c" to define time, then you get
well-behaved, Lorentz transformations, and Maxwell's equations stay sane and "nice".
4) So most physicists choose 3), even though your "Universal Time" definition is possible,
and transformed UT EM fields are (possible but) REALLY NASTY!
5) So it comes down to just, "how do you want to define time?"
a ) Universal? --> real nasty transformed EM equations
b ) Lorentz? --> nice, well behaved transformed EM equations
6) Most physicists pick 5b) It is just that simple.
You say, "this does not prove that time slows down at speed...."
Well, you really do not "prove" anything. You define.
If you define time as "light-travel-distance-divided-by-c" rather than
"Universal Time", then Maxwell's equations are nice and sane,
and you get to use the Lorentz transformations (if you recall,
Lorentz knew nothing about his "Lorentz Transformations" other
than the fact that they left Maxwell's equations intact after transformation!!!)
So, SR is indeed self-consistent, even though you may not like "time dilation" and
other effects with time defined using light instead of "Universal". But I choose to
use SR because Universal Time is just not practical/usable.