Actually it is a fair question, and even though it is a conceptual and basic question, it is well stated and precisely written. It makes clear what you are confused about. It's two factors, one is thinking of active transformations instead of passive ones (see below)[and the two are equivalent once understood], and the other one is, thinki g of the transformations, at least for G, as a continuously rotating body.
First, the most obvious, the rotation G is not rotational motion (which would have you in a non-inertial reference frame, as you indicated). It is instead a simple rotation of the axis of reference, where your motion is described in the statically defined rotated frame. Ie, you can draw the reference axis with any direction being the x axis, and just rigidly draw the other two so that it is a rotation from my x, y and z axis.
[hopefully not to confuse you, but what I described is a passive rotation, i.e., the reference frames, and what you described was active instead, but also continui got rotate. You could have just rotated the body and anything else in your system, a single rotation and not a continuous one, and still describe it as inertial]
The second question of why and what is s, also shows the same confusion. Think of it as not going to the past or future, but recalibrate get your clocks (or calendars) so that t=0 in the new clock is t=-s in the old one. That's a passive transformation. You can make it active by considering t=s in the new clock as t=0 in the old clock (coordinate system with that clock; or alternatively do t-s).
So the best way to get in-confused is to think of coordinate transformation for a new coordinate frame (=system).
For a continuous rotation you'd need the elements of G to be functions of t, where instead of a fixed rotation by $\theta$ It would have been something like $\omega t$.
Once you understand the passive coordinate transformation it's not too hard conceptually to consider them active, they are equivalent.
Hope this helps, and I can assure you that others have had the same silly confusion.