To answer your questions " We are always thought that the West wanted to bring back the Roman empire and they missed it a lot, but how can you say that and think about invading or destroying the Eastern part of it?"

The idea that the Western states wanted to restore the Roman Empire is a complex one. It's true that the Western states did idealize the Roman Empire and sought to emulate its political and cultural achievements. However, their actions towards the Byzantine Empire, the Eastern part of the Roman Empire, were often motivated by more practical considerations such as political power, economic gain and religious differences. The Byzantine Empire was seen as a rival to the Western states, and their efforts to invade or conquer it were driven by the desire to expand their territories and influence. This does not mean that the Western states did not acknowledge the Eastern Roman Empire as the true descendants of Rome, it's more that their actions were driven by other factors.

Additionally, the Eastern and Western parts of the Roman Empire had grown increasingly distinct over time, with different cultures, languages, and religions, which further contributed to the Western states' view of the Byzantine Empire as a separate entity. The idea of "restoring" the Roman Empire likely referred more to the idea of re-establishing a powerful, centralized state in the Western parts of the former empire. The actions of the western states towards the Byzantine Empire were driven by a combination of practical considerations and the idealized image of the Roman Empire, rather than a genuine desire to restore the Eastern Roman Empire as it was.