ELi5: If it's true that "dark energy" doesn't exist, and that the expansion of the universe we observe is just time dilation from the Milky Way's gravitational force, then is the universe actually expanding? And is "heat death" still the outcome of the universe?

There has been a recent scientific observation that suggests that our current model of the universe (that it started with a big bang, and has had accelerating expansion ever since) has been a mistake. I am no physicist, but my understanding is that time dilation from within our galaxy has caused our perception of time to move 35% slower than for the void of the universe. The rest of the universe, moving at "normal" time, therefore appear to us to be accelerating away from us.

https://www.youtube.com/watch?v=hE_xLGgZzFI

My questions- is this a correct understanding of this theory? And what are the implications for the fate of the universe, is it still expanding? Will heat death still be the ultimate outcome?