This is due to the way you saved the component and then imported it. When you imported the component into your new model, it was essentially double wrapped and the outer wrapper is not dynamic. You could see that if you import the component into another file and look at it in Outliner.
Instead of using saving it with File>save and importing it, right click on the component and use Save as... Then save it to a directory you can use for a local collection. When you want to use the component, go to the Components window, select that local collection and drag the component in from there. To make the local collection available in the Components window, click on the Details menu and choose Add collection to favorites.