It is important not to conflate writing systems and languages, because the question appears to be comparing apples to oranges. The Chinese writing system may have remained quite consistent, but so too has the Latin alphabet for instance. Meanwhile, spoken Chinese languages and English are both very different now than 1000 years ago.

But as for the deciphering of texts, Kissinger is talking nonsense. For one, these ancient inscriptions were written in an old form of Classical Chinese, which remained continuously in use as a standard for writing well into the Qing period, but which has undergone subtle changes as well as being greatly divergent from Chinese vernacular languages. If a modern Chinese person can read a 1st millennium BCE inscription, it is not because Chinese languages have been continuous, but rather because they may have been taught a certain amount of Classical Chinese in school. This would be not unlike Latin remaining as a language of the educated in Western Europe, or the many revivals in use of Attic Greek in the eastern part of the Roman Empire, from Arrian in the 2nd century to Anna Komnene in the 12th. But as noted literary Chinese was not a static form, just as Classical Latin was not identical to Church Latin.

For another, the modern Chinese writing system derives mostly from orthographic reforms during the Han period, postdating Confucius by over two and a half centuries, and so Confucius-era inscriptions are far from easily recognisable. An inscription like this from the 800s BCE would not be recognisable to someone used to reading modern Chinese writing.

This answer by /u/keyilan goes into more detail on developments in Chinese writing standards and vernaculars than I ever could; keyilan and I actually weighed in (well, they weighed in with far better understanding than I had or have) on a similar sort of discussion here.

https://www.reddit.com/r/AskHistorians/comments/7xz3b7/are_ancient_chinese_characters_still_legible_for/?utm_medium=android_app&utm_source=share