(I’m not completely sure this is what you’re looking for, but):

Did it ever truly become unimportant? We can all agree that in the very early days (i.e. Hatti, Luwians) it was one of the greatest powers in the Late Bronze Age world. And, as you said, the influential Phrygian and Lydian societies followed after (I don’t know what you mean by “Pontus”, that’s a geographical term).

What followed after was the Achaemenid Persian Empire, then much later Alexander’s Empire, then of course, the Romans. Constantinople became the capital of the Roman Empire under the Roman emperor Constantine. So, we can agree that Anatolia didn’t truly become unimportant before the Romans. Then, of course, we see the Byzantines. Later, the Ottomans.

Anatolia has always been a powerful region, and still is. It wasn’t only the Hittites, Lydians, and Phrygians that were important, also the Byzantines and Ottomans. I would even argue the Ottomans and Byzantines were more influential than, for example the Phrygians.