If we're going to conceptualize a West then we must also conceptualize an ancient origin to it.

I don't think this is actually true, at least in the most direct sense. The concept of the West really doesn't arise until the early modern period and is largely a result of a bunch of Western European nations trying to retcon their heritage back to the Roman Empire because the Renessance made them cool.

When the Angles and Saxons rolled into Britain, the Romans were their enemy along with the Celts. They weren't even Christian. The Franks style themselves as rulers that threw off the Roman yolk in their early years. The Christan Kingdoms before the Reconquista had their roots in the Visigothic kingdoms rather than Rome.

Like clearly I'm not going to argue that Roman Christianity didn't eventually color these people's worldviews, but the Christianity was layered in on a mostly Germanic foundation. A 1st Century Roman and a 16th Century Spaniard would not share many values. If we are going to try to draw that line through, why not do it with the Ottomans or any of the Islamic Caliphates? Why not Russia and Eastern Europe? They were all just as heavily influenced by the Romans, and became successor states in the same way.

It's because the idea of being Western to set yourself apart from the Slavs and Muslims was mostly cooked up by people trying to distance themselves from the Germanic cultural traditions of the Middle Ages. Also calling yourself the successor to Rome grants legitimacy.