"Higher taxes" is a very broad term and so the original article can be right that wealthy people support higher taxes and your "...opposed to higher taxes" observation can also be true.
The incentive for wealthy people is to avoid changes to the status quo. So billionaires tacitly benefit from stiff taxes on millionaires or new billionaires, because then they can't accumulate wealth so easily. Millionaires benefit from stiff taxes on the middle classes and new millionaires because then the competition can't climb out of the crab basket.
If the taxes were going to reshuffle the social order I suspect there'd be a lot less support from very wealthy people. It is a very rare soul who smiles when faced with an actual decline in their social standing.