This is only a partial answer, but I wanted to note and describe the theoretical literature which will come to mind for many research economists when observing (1), "eliminating the FDIC." The simple framework which is often first used to theoretically discuss this kind of question is the so-called Diamond–Dybvig model, which tries to set up a very simple theoretical world in which maturity transformation occurs in a straightforward way. (Maturity transformation is defined nicely in this interview, which I shamelessly lifted from the first reference on the current version of the [wikipedia page] :) Other quick references are a bit dry unfortunately.) There's something of an instability in maturity transformation which can lead to to-called bank runs; Diamond–Dybvig ("DD") try to capture this dynamic in a succinct model.
I'll be careful to note that this model itself is very abstract -- the true financial system is very complicated. Regardless, it has been used as a framework for exploring theoretical ideas related to banking stability for a while.
I'll start with the intuition for the model first. In the model, banks function by converting short-term accounts (savings accounts, perhaps money market accounts) into long-term loans. Banks take depositor money, retain a certain amount, and lend out a certain amount as long-term house loans, car loans, or small business loans. This means everyone gets something at the end of the day -- depositors get a safe place to keep their money with some modest return on savings; homeowners and business owners can smooth out the large costs of housing and capital over many years, and everyone gets a little bit of the slice of "added production" which occurs because the loans could be made. There's some investment projects which will fail (houses defaulted, businesses going bankrupt), but that can be planned for appropriately by the bank.
Not all depositors need their money at the same time -- after all, that's why you put it in a safe spot to begin with. The bank knows (or guesses) how many people on average will need to access their funds, and holds on to that much (plus extra for a "safety buffer"). Everything else is lent out to achieve the above-mentioned benefits.
This system works fine unless, for some reason, enough people start to believe that enough other people will want to withdraw their funds more frequently than usual -- perhaps all at once. Everyone generally understands the structure of the game: stage 1, if there are too many people who want to withdraw money, the bank will fail. Stage 2, if everyone realizes this, they will want to get their money first. Stage 3, I personally need to get my money out before anyone else does, and I know everyone else knows this, so now it's a race.
Now this only happens if everyone believes that everyone else believes the bank will not have enough money for everyone. For a bank "standing alone" this is always a danger. However it also demonstrates a way to nip everything in the bud: institute an insurance scheme for individual depositors. You need to be able to credibly tell the borrowers that, "don't worry, even if the bank collapses, your deposits are insured and you'll get at least X back." This promise itself (and it must be a "good" promise; a promise no one trusts will be useless) will prevent many possible instances of bank runs which might occur when there is nothing intrinsically wrong with the bank.
The theoretical illustration of the above points is the well-known Diamond–Dybvig model [wiki page here, which is actually quite good]. The model is really quite beautiful in laying out the constraints faced by everyone in the world -- depositors, banks, loan-takers. It does so in a very clean and clear way, which is one reason it is so popular. If you're ever deeply interested in these things, I'd definitely recommend working through the model at some point, ideally with someone you can "ping" for input (perhaps here, later). It's a great example of a framework one can use to tackle bigger questions (or alternatively, a great example of how to frame an observed phenomena in the simplest way possible, which is always how you want to start).
This is one of those nice theoretical results which is so intuitive and easy to understand that it can be understood by anyone. In fact, it is literally the main plot device for the classic "It's a Wonderful Life," where [SPOILER!] the first bank run in the movie is quelled with the "insuring" of depositors' accounts with the couple's honeymoon fund. (The real-world equivalent of the honeymoon fund, of course, being the FDIC in the US.)
NOW, all that said, I have no idea what the context of point (1), "eliminate the FDIC," in The Bankers' New Clothes. I suspect the authors are aware of everything outlined above, and they may perhaps have an alternative suggestion for how to handle the "bank runs" problem (aka the "Wonderful Life" problem) which emerges naturally without some deposit insurance system. If they are advocating a "clean sweep" of depositor insurance, that would be very extreme -- but I don't know if that is what they are actually advocating.
There is an enormous empirical literature on bank runs -- see @Lumi's response, particularly the list of bank runs. I'm certain that if you were to choose a bank run from that list at random and search for it in Google Scholar, you would find plenty of empirical papers. (I'm not an expert on this literature so unfortunately cannot answer this more specifically.)