I'm publishing here an important note by John Uebersax, explaining carefully why the translation "Thy Kingdom come" might not be proper for the Greek word Βασιλεία (Vasileia):

"There is an important issue with the English language version of the Lord's Prayer. Specifically, the phrase "Thy Kingdom come" might be more accurately given as "Thy Reign come." Alternatively, "Rule", "Kingship," "Dominion," or "Sovereignty" are arguably better translations of the Greek word here, which is Basileia.

There is a major difference between "Kingdom" and "Reign" or "Rule." The former is a thing, a place; the latter imply an action or process. What we are praying for, in particular, is that God will govern our will and soul; that we are morally purified, cleansed of egoism, so that God reigns. The word "Kingdom" has this psychological meaning only obliquely.

Actually I think both meanings are implied by Basileia, but "Kingdom" loses the important psychological meaning.

A few minutes after writing the above, I found the following confirmation in note to Gregory of Nyssa's sermons on the Lord's Prayer:

"Basileia, the word for kingdom is the same as that for kingship in Greek. The argumentation from "Thy Kingdom come" to the sovereignty of the Holy Spirit can therefore not be adequately reproduced in English, as it depends on the double sense of the one Greek term." (Graef, 1954, n68, p. 187)"

I agree that you have a valid point, George.

My source material states:
"From A Public Domain lexicon by John Jeffrey Dodson (2010). Created for The Greek New Testament for Beginning Readers: The Byzantine Greek Text & Verb Parsing. Provided by biblicalhumanities.org.
kingship, sovereignty, authority, rule, especially of God, both in the world, and in the hearts of men; hence: kingdom, in the concrete sense."

Dodson 'blurs' things a little, as if the meaning of divine reign were that God is the "Lord" and man a slave, not a son and a friend.