good question. and it's true. everything you say is true.

the difference is that our trajectories (which get tweaked by curvature) aren't in space, they're in spacetime.

Imagine an ant and a motorbike. both of them travel the length of a 1 km road. the motorbike goes quickly, and the ant goes slowly.

In space, they travelled the same trajectory.

in spaceTIME, they do not. they might have started at the same place and time, but the motorbike ended up at the end of the road in 5 minutes, and it took the ant 5 days to arrive at the same place. You can draw their trajectories on a piece of paper if you want: label the vertical axis "time" and the horizontal axis "space". they travel the same distance, but it takes less time for the motorbike to cross it, so the motorbike's spacetime trajectory will be steeper than the ant's.