Indeed, the regulations require the opposite of what you suggest: banks are are required to maintain a certain level of capital reserves to avoid collapse even though they would prefer to spend all of it on investments or making loans.

Banks lend more dollars than they borrow from the Fed (or depositors). They don't 'get to keep' money they borrow from the Fed, they repay it at the base interest rate.

The hoarding obviously isn't in cash, just look at their financial reports to know what they do with the money, and it largely isn't lending to businesses for expanding production. Since deregulation, much of the new money has gone to buying stocks. Not exactly helping the economy. As long as the stock market appreciates faster than the prime rate, it makes sense to borrow money from the fed to buy stocks. Especially if the fed is buying stocks too, keeping the market in perpetual inflation driven growth.