No, if rich people want to even stay at their same level of wealth, they have to move money around, spend it, invest it, etc.
I think everybody that makes the claim that the wealthy horde wealth (and I’m not one of them) don’t think that the rich keep stacks of cash under a mattress. The assumption is that the wealth is “kept” in investments that return more than the rate of inflation. (It’s not a high bar, any mildly diversified portfolio does that easily. Even just putting things 100% into an index fund probably accomplishes it.)
If you inherited a million dollars in 2000, you would need an extra $721,000 to make the same purchases now.
Government punishes saving.
However, a million dollars invested in 2000 would be worth over $4M today if simply invested in a Dow-tracking index fund. That’s far ahead of inflation, that’s more than doubling your real wealth.