Why is King Arthur considered an icon of England if he was a Romanicized Celtic man who fought the invading Anglo-Saxons?

While there is certainly questions about how King Arthur was or was not based on and whether or not he was real (for my money, he may as well be), the real question is the status of the Celts, Rome, and the Anglo Saxons in the English imagination. The modern English (or British, and I understand there is a bit of socio-political positioning involved in the distinction) is, like all identities, a constructed one. That means that while there was an ancient place called England and genetic continuity and the like, the meaning of the identities are created in the present and justified with historical narratives. The artificiality of this process often lends itself to debate and confusion (see how France and Germany both claim Charlemagne), which in England took the form of a nineteenth century controversy over whether England's essence was Celtic, Roman, or Anglo-Saxon. As these debates always do, the debate was less about the past (it is somewhat incoherent, historically speaking) than the present: the aristocrats liked to point to the Romans and the Victorian imperial project was obsessed with Rome, but it was somewhat politically inconvenient because it represents a foreign conquest. The Celts were liked by those of a Romantic persuasion and allowed an image of fierce independence to be projected, but also had certain inconvenient associations, such as the Irish. The Anglo Saxons emerged as something of a compromise candidate. This is detailed very well in Virginia Hooselitz' Imagining Roman Britain.

But of course in reality these sorts of debates are irrelevant to how the past is used in a popular context. Most people today accept Hadrian's Wall as a sort of symbolic border between England and Scotland, and there is a giant statue of Boudicca in London, and nobody finds this contradictory. Ultimately these scholarly debates are not only imposed on the past but also on the popular imagination.

So, in short, yes you are right that it doesn't make sense, but it is a good story so nobody cares.