If they were allowed to fail the economy would take a hit, not only in the US but worldwide. However, humanity is very creative, and adaptive, and I think you'd find that new credit companies would form, granted not to the size and scope of groups like citi, but would certainly help ease the economic turmoil until new companies grew to the point of being able to provide stabilization.
I don't think you understand that "the economy would take a hit" really entails.
The problem is that "credit" is not just used to spend money you don't have (although that's certainly one of the causes of the current crisis). It's also used to spend money that you DO have, but don't want to be inconvenienced by carrying around. As a simple example, I can't remember the last time I used a traveler's check. My ATM card, which is also a VISA card, works almost everywhere and in every currency; I can travel around the entire world and buy things
for which I already have the money in my bank account.
If the VISA company suddenly collapsed, I wouldn't be able to do that. But I also wouldn't be able to use traveler's checks, because almost no one issues them any more, and almost no one accepts them any more. (The few places that do are generally banks, which would have collapsed along with the VISA company). Furthermore, I'd have a much more difficult time because the currency conversions aren't automatic -- if I had checks in dollars, I couldn't convert them to euros without going to a bank, and the bank relies on the existence of credit to do the currency exchange.
The systems that we had in place in the 1920s to handle this problem, for the most place, no longer exist. The system we have in place today to handle this problem assumes the availability of cheap and widely available credit. Eliminate the credit and there are
no systems for large-scale monetary transactions -- or even medium-scale monetary transactions, or small-scale transactions across long-distances.
It makes no sense, for example, to eliminate the entire hospitality industry
because we don't want to prop up the banking industry. But that would happen if the financial system collapsed, precisely because innkeepers would no longer be able to accept payment in any form other than cash -- and there's nowhere near enough cash in circulation to make the industry viable.
Yes, in the long run, "there will once again be lemon-soaked paper napkins." But the transition would be ugly, and go far beyond the "mere" billions of TARP funding.