If you want to tax the wealthy, then simplify the tax code and remove loopholes and deductions. Leave rates alone.
Your calculations was right in that idealistic tax system, but the real world is really messy with lots of exceptions, Tax credits, Benefits, and such.
Building on your example, I will add a little Child Benefit to make a slightly less ideal idealistic-scenario.
Assume If earning < 100: child_credit = 3
Assume 1 Child
If earning 99: take_home = (99 * 0.9) + 3 = 92.1
If earning 100: take_home = (100 * 0.9 + 1 * 0.8) = 90.8
-----
I chose this example because I remember vividly a story about a family in the UK that avoided getting promoted because their take-home would decrease. Of course it have to be a very small promotion in order to not be worth it. I don't know if cases like this is rare in the US, but it is generally a thing.
https://www.zerohedge.com/news/2012-11-27/when-work-punished...
Further, a flat wealth tax has immense inefficiency in that it forces people with $100s or $1000s of dollars to their name to calculate their wealth for a $1-$100 payout to the government rather than do something productive with their time.