Psalm 24 is, as you say, a likely source for this Beatitude. So is Psalm 73:1, “Surely God is good to Israel, to the pure in heart.” And I agree with your take on “heart” as the seat of will. We are not talking about feelings here at all. I am curious, though, as to what it means to “see God.”
I am a panentheist. I believe that all (pan) reality is in (en) God (theos). This differs slightly from a pantheist who equates God with the universe. While I believe the universe is part of God, I do not believe it is all of God. God is not other than the universe, but God is greater than the universe.
Anyway, as a panentheist, I see God all the time. Everything I see is God. Of course this is an intellectual seeing, and not the transformative seeing that reveals the kingdom of God. To achieve that level of seeing one must be “pure of heart,” so it is vital to our understanding of this Beatitude that we understand what being “pure of heart” means.
Let’s start with the word pure, bar in Hebrew. When coupled with the word lev, heart, it means “straightforward;” one who is pure of heart is one who is simple, honest, without deceit and conceit. Now, why would this person see God?
To me the answer is this: the simple person sees self and others without labels, just as they are in and of themselves, which means as manifestations of God. It is only when we drape people in “isms” and ideologies, seeing race, religion, creed, ethnicity, and political affiliation instead of the person they are that we fail to see people as God.
When we do see people as they are—God manifest in time and space—then we see God in, with, and as all people (and I would say all beings as well). This kind of seeing is transformative in that it leads to a deep sense of compassion for all beings; we see our neighbor (mineral, vegetable, animal, human, etc.) as our selves. This awakening to the nonduality of God in, with, and as all reality is the key to living the kingdom of God.
And to do this we have to see with the blinders of belief.
I think this leads easily into your comment on the divided heart (as opposed to the simple or whole heart). A divided heart separates self from others, from nature, and from God. But division has to be learned. So much of what we do as parents, clergy, and teachers is to cloud the heart, distort the eye, and promote and perpetuate the delusion of division and duality that excuses our desires to exploit and control the world and those in it.
I would even go so far as to say that by saying, “Seek you first the kingdom of God, and all these things shall be added unto you,” (Matthew 6:33) Jesus is challenging us to step beyond belief, to cleanse the heart of division, and, in so doing to then see the world as it is: the kingdom of God.