Trickle down is a political label used by people to attack policies they don't like by claiming a policy helps wealthier people at the expense of less wealthy people.
But that's not the mechanism we're talking about with housing at all.
What we're talking about with housing is that there are 5 houses and 10 people, and in that kind of scarcity environment, 5 people are going to lose out. In most cases, it's going to be the 5 people with the least money.
But even if you do a lottery or something else, 5 people are still going to lose out and that's bad.
Build more housing.
As the wikipedia article says, "trickle down" is a political term. Any proposed tax policy deserves real debate on the pros and cons - using political terms like "trickle down" (or for that matter terms like "tax and spend", "bleeding heart" etc.) are not done to help advance the debate. They aren't trying to advance an argument - they are trying to shut down debate. People who use political terms like that are usually trying to either poison the well or are creating a straw man.