Any rational investor isn't going to be aiming to cover inflation, they are going to be trying to maximize their rate of return. It wouldn't matter if the interest rate was above or below the inflation rate, investors will always shop around to find the highest return they can get.

I think "greed" is an unhelpful term as it is too emotionally charged for what is really just rational behaviour given economic incentives. So I would avoid calling banks greedy for trying to maximize profit by offering low interest rates just as I would avoid calling consumers greedy for choosing the bank that gives them the highest interest rate.