Did the group contain every single nodes of your scene? Did the hierarchy of your joints change during the process?
When you group an object, it sort of create a new scene for what’s inside the group. This way if you move the group around, scale or rotate it, it doesn’t change the coordinates and values on the objects themselves inside the group. Think of it as a box and everything inside is relative to that box, just as ungrouped objects are relative to the whole scene.
Another problem I see in your scene is that your object doesn’t have enough polygons. The way animation works is that when you deform something, it calculates the new value of each vertex depending on their initial position. If your object only has vertices on its corners, there can’t be any curve in the object since there are no vertices in the middle to be calculated after the deformation.
If you’re familiar with vector drawing, you’ll notice that working with Maya is similar but on a 3D plane. You place points and the application creates a shape from it.
Try using the “Add Edge Loop” tool under “Edit Mesh” to give it more divisions. Do it both vertically and horizontally. The more divisions you have, the smoother it will look when it curves.
This is why the bottle has so many “lines”, or edges properly said, in the images of the tutorial.