There is indeed time dilation. But as Feynman points out, the theory says time can't stretch without space also stretching. So how does time dilation help you understand the apparent attractive force of gravity? Students scratch their heads because it doesn't really. A good explanation of gravity talks about the space part.

By the way, be careful when you say "expansion of the earth" - skeptics might think you're promoting a nutty new theory. "Expansion of the space that contains the earth" is more on target for Einstein's theory.

Answering your question: I think this take on spacetime curvature calls into question many assumptions about "the expansion of the entire universe". Space expands at different rates depending on how much matter is nearby! Be skeptical of theories that make big pronouncements about The Universe without even acknowledging the easily measured expansion of space around our own planet.