I understand the concept of spacetime being warped (I’ve seen the rubber sheet analogy a thousand times), but does that mean that everything is *technically* always traveling in a straight—albeit warped—line?
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Yes. Without outside forces, things move in a “straight line”. I put that in quotes because it is like if I asked you to paint a straight line on a curvy boulder. It’s a little bit ambiguous what straight means when the world is curved.
Try not to think of gravity as something that forces the universe to curve. Think of gravity as evidence that the universe is curved.
Imagine that the Earth was perfectly smooth and that you and a friend were standing side by side along the equator. Look around. The world looks flat doesn’t it?
Now both of you face north and start walking. If the world was truly flat, you guys would walk side by side forever. Instead, you two will eventually come together as you approach the North Pole. Nothing is pushing you together. You naturally come together because the world you’re in is curved even though it appears flat. How soon you come together depend on how curved your world is. E.g. the Moon has a smaller radius so it is more curved and you and your friend wouldn’t have to walk as far to see yourselves come together.
That’s what gravity is. It’s not a force that pulls things together. It is evidence that the true shape of the world is curved.
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