It's a little bit tricky conceptually, but the key word here is relative. Time distortion only exists relative to another observer (or in other words from a particular frame of reference).Yeah, the galaxy is moving at 1.3 million mph (relative to background radiation) but we're part of the galaxy (and moving along with it) so it's not distorted for us because we're moving at the same speed.
To say it distorts time you'd have to compare it to something that wasn't part of the galaxy (and was moving at a different speed), such as from another galaxy. Basically you'd have to measure time from a frame of reference that we don't have access to (since, again, we're part of the galaxy and are moving along with it).
Or another way to think of it is this. Going back to the jet example, if you were born on the jet going around the world, spent your entire life there, and will never have any way to escape or get information from outside of the jet, then it's meaningless to talk about how fast time passes on the jet compared to Earth because you can never measure time from Earth.