I would suggest this understanding does not reflect the nature of God. People are under the impression that everyone would want to go to heaven and that God somehow forbid people to do that. The conclusion would be just as you've portrayed it.
Well, what would happen if instead people really want to go to hell? We know God and what He is like but instead of wanting Him, we want to go to hell. In this case God would be on a rescue mission to save some of His people. He would be dragging us out of Egypt. This would certainly paint God in a different light.
I would suggest that Romans 1-3 paint people exactly as wanting to go the hell. They don't seek after God. Some may say why did God create man this way, but Paul answers this with his potter-clay analogy. We cannot presume to know the mind of God except that in everything it is right, holy, and just-filled with love.
we have free will, we make choices
You cannot make choices until you are set free by the Son. Otherwise, what exactly are you being set free from?
As one poster put it, if you can't put "Jesus Loves Me." on a t-shirt to distribute, your doctrine has gone off the rails somewhere.
I would have loved to see Jesus handing out those t-shirts to the Pharisees who He called "sons of the devil" or the "brood of vipers" crowd.
Without "synergism," if this is what you mean, then man is removed from the equation. There is no relationship with robots.
The relationship begins once God opens the eyes of a person and helps them to hear the gospel.
Everything good that comes to us or through us is from God, but we can say no.
I think its a bit persumptious to think we can say no to God. Jonah tried it and it didn't work. If you think I'm wrong, just try it yourself.
And that is predetermined, and you cannot choose otherwise before, your choices are predetermined afterward.
This is not human choice, by any stretch of the word.
I am of course describing the view according to Westminster Confession.
People do it every day, you likely included, all over the world.
And in your theology, this desire for hell would have been implanted by God? That's what it sounds like to me.
I agree with most of your post. I primarily disagree on double predestination.
I wouldn't say that all want to go to hell. We want what we want when we want it, and the result is hell. On earth. Call it God's subtle direction. Unless we repent and believe. Our difference, primarily, is I believe this is a free will choice.
As a sidebar, I don't think we can fully conceive what heaven and hell really are. I don't know if you've read Dante's Inferno, but it is a thought-provoking work postulating that Hell is where the unsaved only get what they really want.