Is wealth inequality actually a problem
I think that you are more or less than correct in thinking that cash that is never spent (nor expected to be) may as well not exist. Thought experiment: suppose that, unbeknownst to you and everyone else, £1 million suddenly appears under your mattress. If you never discover the money, it is hard to see how it could change anything in a meaningful way.

There are some caveats, however. First, a game theoretic point: even if rich people never spend their money, threats to spend their money might influence outcomes. For example, they could pledge to donate lots of money to the political opposition unless the government does what they say. If this threat works, they will never need to pay out.

Second, a ‘non-economic’ point: money gives you options, and people might enjoy having options in itself (even if they don’t spend the money). One might object to the kind of ‘options inequality’ brought about by an unequal wealth distribution even if rich people never spend their cash.

Third, to bring us back to the real world, we should stress that, usually, rich people do consume more than poorer people. So while it might be interesting to discuss wealth inequality without consumption inequality, this is not a phenomenon which we are actually facing today.