>>"A bank that only writes loans and doesn’t have deposits is the opposite of robust."
Maybe they are, but modern private banks already do that.
Before a bank concede a credit it's just checking if it's a good business, not if it has enough deposits or enough reserves.
When giving a credit, banks, are, basically, creating money that it's destroyed when the debt is payed.
A posteriori of giving the credit, the bank will search (assuming it have not already enough) the legal reserves requirement for the new created money.
Note, that this is the opposite of what we normal hear. In reality, first come the credit, then the reserves.
The bank that gave the credit, trying to meet its legal obligations of reserves, will try to get reserves loaned from other banks, creating demand for reserves in the inter-bank system. If the Central Bank does nothing, the added demand would increase the interest rate.
Modern Central banks don't try to control the quantity of money in the system, but the interest rate. If the are not enough reserves for the existing demand, in order to keep the interest rate in their target, the central bank will increase the reserves in the system.
So, deposits are not necessary, otherwise, the quantitative easing programs of the last years, would be impossible.