there are those who say that the Scone stone is actually Lia Fail, brought over by the Irish settlers. Leaving the invaders aside there has only ever been one king of the whole of Ireland, Brian Boroimhe [pronounced Boru], as the King of All Ireland didn't rule the northern corner.
Regarding the Welsh kingdoms mentioned these are really sub-kingdoms of the South Welsh as historically you also had the North Welsh [last known as Yr Golledd in what is now The Borders] of northern Britain and the West Welsh in what we still cal the West Country. The East Anglian knigs it is now thought could have been Britons, giving us an East Welsh lost to the Romans. On analogy with Erin there may [according to me alone AFAIK] have been a central Wales from which overlords such as Vortigernos and Arthur came.