"If that [$10bn] money supply was added directly to the bank accounts of our bottom third, that would be $900/month...Ok so what is wrong with this naive idea?"
There is a bug in the construction of the idea; it's quite common, and even gets exploited. This post is to present clarity on that point of confusion.
There are tiers to measurement. The first is a count of something, such as 10bn dollars. The second is a rate of something, such as $900/month. (The third would be an acceleration of something but that doesn't come up in this problem)
It's not a little oops, I didn't mean to do that when the toll booth on the road keeps taking money to pay for the road long after the cost has been recouped. So clarity is a good idea. Here's some examples across differing fields so that the specifics can be seen:
Level-1: I drove 25 miles. I used one gallon of gas.
Level-2: I used gas at a rate of 25 miles/gallon
Level-1: I made a bajillion dollars. It took a year.
Level-2: I made money at a rate of a bajillion/12 dollars/month.
Level-1: I wrote ten essays. It took all week.
Level-2: I wrote essays at a rate of 10/7 essays/day.
And so back around to the original problem, with a fictional fill-in
Level-1: They got 10billion. ***missing part*** It was distributed over five years.
Level-2: They received money at a rate of 10bn/(5*12) dollars/month over the five-year period.
Don't feel stupid; I suspect this conceptual hiccup is a product of the combination of homo sapiens brains and the way things are most often taught.