I rarely see this spelled out, but in the more modern thinking, the working class are "creatures of flows", and the capitalist class are "creatures of stocks": i.e. the latter has meaningful assets, but the former subsists on wages.
Obviously this is just an approximation, but "class is calculus" is a very useful mental model.
By that ones, yes, the vast majority of people have negligible liquid savings. So the interest rate doesn't directly matter. Yet, the rich also "own the means of production" --- i.e. real assets, which also doesn't get inflated away. So it doesn't obviously matter for them. So what does it effect?
The essential original Keynesian insight (what most of the factions since might agree on), is that since the point of owning things in Capitalism as money, even if you aren't holding cash, the interest rate effects how easy it is to invest, but also what investments are productive w.r.t doing nothing. This spooky shit is what is supposed to give monetary policy it's power.
The differences in opinion in those factions come down to whether there is always enough to invest in, and access to credit is the limiting factor, or not. The Post-Keysnians would say relying on monetary policy alone is stupid, because (to distill in a parable) if no one can buy your stuff, it doesn't matter how cheap the factor is it's not worth it. That's when directly employing people (jobs guarantee) or handing out helicopter money can make a difference --- it's "root" demand, not some spooky shit effecting supply.
Back to the calculus stuff, an interesting idea is https://en.wikipedia.org/wiki/Demurrage. This is "inflation for stocks not flows". In short, we want money at rest to go bad, so rich people and instutions are forced to spend and we don't grind to a halt via the paradox of thrift. But we don't want people's wages to inflate away, because they certainly cannot bargain hard enough to get enough raise to keep up. Demurrage is wonderfully hamfisted in trying to do the former but not the latter --- money at rest becomes less money, money flows retain their purchasing power.