It helps to remember that each design in Fusion 350 is it's own component with it's own origin.
When an assembly is imported the symbol of the browser root changes to indicate that it is now a component group. Each individual component has its own origin, but the group itself also has its own origin that lives at the top level.
If there are no changes to the assembly I don't usually turn on the timeline and keep the assembly in direct modeling mode. This uses less data and helps keeping assemblies lighter from a data standpoint. In this case we want to re-orient the parts in reference to the top level origin.
To achieve that I create a rigid group of all the components by selecting the first component in the browser and shift-selecting the last component and then rigid group those.
What the rigid group joint does is simply freezes the origins of all the selected components in reference to each other. This way the assembly as a whole can still be moved in reference to the top level origin, because I did not include the root of the browser (the top level component) in the selection.
Now I can simply align the geometry with the origin and use the move tool to rotate the assembly to my liking. Make sure you select "Component" when sing the move tool.
When everything is in the desired location and orientation, I delete the rigid group and now select the root of the browser and create a rigid group joint that will now include the top level origin.
Save. Close. Done!
When that design is now inserted into another one it will behave as expected and the assembly should not "explode" as shown in your screencast.