I seem to recall some Investment Banks going belly-up, and some Insurance companies as well. ING was forced to sell their Canadian operations to generate cash (sold to CIBC) and shut down entirely in the US at the same time, so it traversed borders.
But I want to address why Mortgage Companies and consumer Banks could not be allowed to fail. You have to start with how banking works. I will try to keep it brief, but it's complex.
Banks reconcile deposits, loans, and withdrawls with the Central Bank every night ... in the case of the US, the central bank is the Federal Reserve. The role of the Federal Reserve is to insure the economy has enough money to function, expand at a reasonable rate, but not too fast as that causes inflation. They do this by manipulating certain aspects of Banking.
One aspect of banking they regulate is the amount of deposits required to correspond with the amount of loans they may grant. These days the figure in most countries is around 10%, give or take a few % depending on the economic conditions and the goals to either restrict or expand the money supply ... all the money available in the economy to generate wealth and promote business.
What that means is a bank must have X dollars on deposit before they can loan Y dollars. With a 10% requirement (for example) that means for every $1 million on deposit, they can loan $9 million dollars to business and the public, for a total of $10 million.
When a bank makes a loan, that money is created "out of thin air" at the moment the loan is made. The bank does not have the $9 million, it only has $1 million, but has the ability to loan $9 million.
The moment a bank deposits a loan into an account, or writes a check, that money exists. Prior to the loan, it didn't exist.
The borrower's ability to repay is the real currency of banking. This is fundamental to understanding how banking works.
Overnight, but after the fact of a loan being created, the Federal Reserve then loans the bank the money at the prescribed rate (usually set every week) to reconcile the books. This is sometimes what is meant by the idea that the government can "just print money".*
When depositors remove money from a bank, it has to reduce the loans it has in it's portfolio. Generally this is done via loans to businesses which are often "Demand Loans". A Demand Loan is one the bank can call in at any time.
So, banks will call in loans, demanding the full amount be repaid immediately. This causes businesses to contract, layoff employees, sell assets (and in a recession, at a loss as most buyers are also short of cash).
If this is allowed to continue unabated, it will make the recession more severe and more difficult to recover from. So Government will step in and support Banks in order to minimize the effect throughout the economy. Broadly speaking, they don't have a real choice here, as failure to act will severely cripple the economy as a whole, and lead to greater job losses and increased business bankruptcy.
Similarly with Mortgage Lenders, some forms of mortgage can be called in, if not immediately, then on an annual anniversary date. Panic selling depresses the home resale market.
There is one final aspect to this and that is how a dollar generated by a loan is spent. The old Economics Textbooks said that every dollar is spent repeatedly over a year, generating $7 in wealth for every $1 generated by a loan. When you get a loan, you buy something, that money goes to a second party, if they deposit that money in their bank account, the second bank can now lend it's $9 for ever dollar deposited, that money is spent, and so on. Not all of it is available to be re-used (so to speak) every time, but most of it is, and that results in a net $7 increase in the money supply.
Today, with more electronic transactions available to everyone, business and consumer alike, that ratio may be even higher than in the past.
So, there is a natural aversion to "bail outs of banks" by anyone who pays taxes, since that is where the money comes from (some day ... the US Government operates on future tax revenue, not current, to a large extent). But the cost of not doing so is greater and if left un-addressed, layoffs and bankruptcies would be greater and would touch every citizen versus just "some".
With regard to Investment Banks ... the "Wall Street bailouts" the argument is a little weaker, but only a little. Those entities, if allowed to fail, would take with them the majority of the nation's pension funds, affecting the retirement plans of workers in the nation.
With regard to the Automakers, GM, Ford and Chrysler were all solvent with adequate cash flow, but the banking crisis reduced the ability of banks to lend money to new car buyers. That affected cash flow significantly, making the automakers in danger of insolvency. Some say Ford didn't need help because they were well run, and GM and Chrysler were not. That isn't actually true.
Ford, due to collosal mismangement earlier in the decade**, had already sold off most of it's subsidiaries (Jaguar, etc) and secured high interest rate loans prior to the crisis. So they had money to keep operating. GM and Chrysler didn't need to restructure, but when the banking crisis unfolded, banks could not lend them any money, as they simply didn't have the assets to support new loans. GM actually did go bankrupt ... your "old" GM shares make nice wallpaper today. Both GM and Chrysler paid off their loans with high interest (up to around 20%) and the US Government made money on the deal (as did the Government of Canada and the Province of Ontario, who also lent money to the two).
* Some governments, at times of financial crisis, do resort to "Printing money", typically by offering it to banks at a negative interest rate. Not surprisingly, this is readily accepted, but as the economy as a whole can't produce more overnight, it results in runaway inflation (eg Zimbabwe, venezuela).
** Ford lost $US 12.7 Billion in 2006, an average of just under $2000 for every vehicle sold.