Luke 9:60 seems to me to be Jesus helping the man to see the real situation. Naturally we humans feel sorry for those who have died and want to honor them often with the closure of a good burial.
The words of Jesus
"Let the dead bury their own dead" were intended to convey the real situation of the world. Not only should we be sympathetic to those who physically died, we must realize that without Christ we are in death already.
The Christless life is
"alienated from the life of God" (Eph. 4:18).
"This therefore I say and testify in the Lord, that you no longer walk as the Gentiles also walk in the vanity of their mind.
Being darkened in their understanding, alienated from the life of God because of the ignorance which is in them, because of the hardness of their heart." (Eph. 4:17.18)
Apart from Christ we are estranged from
"the life that is life indeed" which the Christian is exhorted to lay hold of
(1 Tim/ 619) . Christ is the life of God
(John 1:4) because Christ is Jehovah God become a man to be our life as
"the Spirit of life in Christ Jesus" (Rom. 8:2)
The proclaiming of the kingdom of God is living in the kingdom and proclaiming Jesus is both the King of the kingdom and the life of God.
Without being born of God we cannot even see the kingdom of God.
"Jesus answered and said to him, Truly, truly, I say to you, Unless one is born anew, he cannot see the kingdom of God." (John 3:3)
If one just proclaims a kind of outward government of the Old Testament God without receiving Christ into to them for a new birth they really do not see the kingdom of God. For what they see is only outside of them as a administration bossing people around outwardly like a strong government.
John emphasizes the end of the alienation from the life of God by the Spirit imparting this divine life into people as a new birth -
regeneration.
Before His resurrection to proclaim the kingdom of God was primarily proclaiming the need for repentance.