The original note, as written at the time
General relativity predicts that motions moving close to the speed of light experience time slower. Small objects may not be able to harness the amount of energy to pull off such a feat. But if you stretched your vantage point out wider to a galactic level, then it could be possible for an entire galaxy to move at such a speed. Now consider the assumption that the early universe was moving way faster than it is now.
Ongoing development
General relativity predicts that things moving close to the speed of light experience time more slowly. A small object cannot harness the energy to do that — but stretch the vantage point out to galactic scale, and an entire galaxy moving at such a speed becomes possible.
The reframing is the interesting move, and it is a good one: the energy required scales with mass, and so does the available energy if the thing is a galaxy rather than a ship. What is impossible for a kilogram is not impossible for a hundred billion suns.
What this would give you is a clock. A galaxy moving fast relative to us is a galaxy whose internal time runs slow, and if the early universe was moving much faster than it is now, then the deep past may have experienced less time than we infer from the light.
That is a real and respectable question, and it connects to a genuine puzzle: some early structures appear too mature for the time available. A relativistic correction to cosmological time would be one way to buy the missing time.
The honest caveat is that cosmological time is not a simple thing. In an expanding universe, relative motion between distant regions is not the same as velocity through space, and the naive application of special relativity to a whole galaxy is exactly the error that produces bad cosmology.
So the idea is not a theory. It is a prompt to ask a better question: when we say the universe is 13.8 billion years old, whose clock is that, and what could make those clocks disagree.
Ask whose clock. It is usually the right question.