The problem that is being struggled with is that HNWI don't have frequent taxable events, generally because they hold assets and make savings. However there is a perception they aren't providing value (which is mostly wrong).

Generally any scheme you come up with to get at them will also affect everyone else, and to a far greater degree since they have less money and diversity to start with. And the HNWI have far greater incentive to move their money and themselves, as well as pay people to work out how to retain their money. You can move to almost any country in the world by plonking down a lump sum as an "investment" or as savings. (eg see how Dotcom ended up living in New Zealand.)

The usual solution to this is to try tweaking the tax code, seeing what the effects are and repeating. However scumbag politicians and the people who vote for them distort that process, so it doesn't have the intended effect.

An actually representative democracy would be a good first step rather than tax rate changes. Watch this talk about the US - the UK pretty much follows behind a few years later https://www.youtube.com/watch?v=Ik1AK56FtVc