No organization cares about you. Organizations aren't capable of this.
Your bank, certainly, doesn't care. Neither does your HMO or even your car dealer. It's amazing to me that people are surprised to discover this fact.
People, on the other hand, are perfectly capable of caring. It's part of being a human. It's only when organizational demands and regulations get in the way that the caring fades.
If you want to build a caring organization, you need to fill it with caring people and then get out of their way. When your organization punishes people for caring, don't be surprised when people stop caring.
When you free your employees to act like people (as opposed to cogs in a profit-maximizing efficient machine) then the caring can't help but happen.