In the traditional bank model, banks didn't rely on short-term funding.
Basically, they would attract deposits from savers and use these deposits to lend to customers (firms and mortgages). In this situation, banks are self-financing as long as savers don't demand all the money back the bank is financing its own loans.
However, what happened is that banks realised they could borrow money on money markets (short term funding) at a low interest rate. This is because banks were trusted to pay back these short term loans. If they borrowed short-term funds they could then lend out more mortgages and make a higher profit margin.
- e.g. borrow short term funds at 2%, lend mortgages at 5%. - profit margin.
However, in the credit crunch, everything changed, suddenly financial markets woke up to the fact bank liquidity was stretched and actually banks were losing money. Therefore, banks found it very difficult to keep getting these short term loans. Therefore, they struggled to gain enough short term finance.
Why can't or don't banks issue longer-term debt just like other corporations?
They can. For example, banks can issue more shares. But, in the credit crunch it was difficult to raise money in any manner because no body wanted to lend.
I can understand that certain banking institutions (i.e. deposit taking ones) must fund themselves in the repo market to avoid reserve shortfalls, but what about investment banks and non-depositary institutions?
It depends on their strategy. If they have taken risks and lent out more than prudent, they may find themselves with a cash short fall at various times