Could do. That might be captured sufficiently by the gift tax though. A bigger issue could be evasion through establishing trusts and the like.

The U.S. has a "unified" gift and inheritance tax. If you give away $5M while you're alive and die with an estate of $5M you (1) pay no tax on gifts while alive, and (2) your estate pays tax based on an estate size of $10M at the time of your death. There are a few wrinkles (e.g., (1) annual per-person exempt gifts of $14k/person, and (2) you do start paying tax on gifts while you're alive once total gifts pass a certain value), and gifting during lifetime has some advantages (e.g., getting growing assets out of your estate while they have a lower value), but that's generally the way it works. Gift tax and inheritance tax operate on a "unified" basis.

Then you just donate the money to institutions your progeny would just happen to enjoy: your alma mater, the country club, private schools, etc. Segregation would still be maintained through astronomical real estate prices.

Your position is that increasing the estate tax would do nothing to help segregation? Or that it wouldn't do enough and we should also do more?

I'm not confident enough to make a definite prediction, but it seems plausible that it could make some forms of inequality worse. Attempts to level out education, for example, have resulted in the truly wealthy finding other outlets (private schools, expensive zip codes) to create the educational experience they want for their children.

In education we never actually integrated though. A few locations were integrated but only for a little while. It seems more like that the truly wealthy found those outlets regardless of what else was happening in education. It also seems strange to say: we shouldn't tax rich people because somehow, through a plausible method that I can't explain or predict, it will make rich people better off and poor people worse off.

I was referring to public schooling in general, not racial integration specifically. Integration among social classes is arguably getting worse now. You only go to school with the kids of bank vice presidents if you can afford to live in the same neighborhood.