In your video, you are making the problem very complicated. It is better to plan out your animated textures, and have everything start at frame 0 so that the animation and the textures are all synced together.
This problem is easier to solve with some old-school math. You’re going to have to figure out which frame you want the animation to start and then take the first frame and copy it for every frame before that. In other words, if your animation is 1000 frames long but you only need the animated texture to start moving at frame 500, take the first frame and make 500 frames before that. That way the animation plays at the same time as the texture starts, but it plays 500 “static” frames and then you only see it moving, animating at the time you need, from 500 to 1000. Does that make sense?
This way all textures start playing at frame 0 with the main timeline, but you only see them change when you want them to.