don't Welch on the Welsh

This is a long running petpeve of mine- watching a show or reading an article  that states Saint Patrick or King Arthur are English or lived in England. They were neither- they were Welsh.
The English did not arrive in the British isles until the late 5th - early 6th century.
They called the population living in the islands Welsh- which some think have its root in the same word Walhaz which meant stranger or forienger which would become Walach and Welsh.
Saint Patrick was born in the early 5th century- which means even if you pulled up clear evidence he lived was born in London- he was still not English he was Welsh as london was not an English city as the English had not arrived.
King Arthur - if he had any basis in reality would have fought the English invaders. In either case they are to be called any modern nationality it would be Welsh.