I wish people would stop saying this. The "American Dream" historically was only tangentially about a house. The real American Dream is being free to live your life as you see fit, without onerous, intrusive government getting in the way, and being able to go as far as your talents and motivation would carry you. That might mean a house or it might not.
The best way to restore the American Dream is not to sell houses to people who can't pay for them. It is to slash the government back to the proper distance, i.e., remote from the lives of everyday people.
Just think, at the turn of the last century, you could literally go for years without paying any attention to Washington at all, except at election time. Now, wouldn't that be a Dream.
You said it best!
What you're talking about should be a right, a mandate, a demand, or an absolute, as in the beginning of Declaration of Independance "Life, Liberty and the pursuit of Happiness" - not a dream. A dream is not forceful enough of a term for what you are describing there.