That sounds like an interesting theory; is it yours, or do you have any citations / references you could share?
Einstein published his seminal works in German, and we'd be more likely to have E=mk².
1. an object at rest has a world line which points (has tangent vectors) in the time dimension
2. light has proper time (dtau) = 0, so clocks moving at the speed of light dont tick
3. the magnitude^2 of an objects 4-velocity is c^2 (objects move through spacetime at c)
4. light has no 4-velocity (because dtau=0, you cant divide by zero)
You can't say (3) means objects have 4-velocity c in spacetime and light has 3-velocity c in space and so that means the time component of 4-velocity for light is zero.
Because light has no 4-velocity.
Through special relativity, our understanding of mass, gravity, and spacetime are linked. If something has no mass, then special relativity can't describe how gravity affects it's spacetime reference.
Remember, however, that this explanation is based on the mathematics that explain the observations we've made or theorized. Just as the map is not the territory, the math is not the universe.
If you're at rest, you have maximum time velocity (1 you-second per frame-second). If you're at the speed of light, it's zero you-seconds per frame-second.
This is described by the Minkowski space, which is a metric that puts two events the same distance apart in spacetime regardless of reference frame.
Greg Egan's series "Orthogonal" looks into what the universe would look like if time didn't have the opposite sign (so that time is another dimension just like x, y, z). The effects of that one sign change are very wierd.