Long post, so I'll do the TL/DR first:
TL/DR: 1) Fractional reserve banking isn't evil; 2) there is most certainly a role for banks in the future bitcoin economy; 2b) right now, those functions can be done in a P2P environment, but if the bitcoin grows exponentially, the economies of scale will give banks/centralized lending pools huge advantages over the p2p lending marketplace; 3) should all of those happen, then a bitcoin Central Bank should be a welcome addition to the bitcoin economy. All of this depends on what you view the functions of banks and central banks are though, you'll have to read the rest of this to see how i describe it though.
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Now for the lengthy post
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PART ONE:
P2P direct lending and such works in the bitcoin community right now because it is so small. Should bitcoin denominated economy really take off to the extent that it's marketshare against the big boy currencies (USD, EUR, etc) is sizable, peer to peer won't work so well anymore. Someone wants a loan and submits their request to this forum, for instance, which could trigger 1,000 potential lenders each having to perform their own due diligence. Just doesn't work.
So, if at some point in the future, people decide they're ok putting some of their wealth "at risk" (though ideally, the "at risk" shouldn't be any more risky than a dollar deposit in a bank), they could transfer funds to a company who's business is specifically lending funds at interest, performing due diligence before making such loans, etc. People could even specify their liquidity requirements with that company - some funds they want to be able to transfer to their own wallet on demand; since those funds can be requested at any time, the company would be very constrained about what they could do with those funds; other customers might be comfortable taking, say, 5% of their funds and loaning it out for longer; they can earn a greater return, but would be constrained with their liquidity slightly. Maybe they can't access their funds at all for the time period, or maybe they can redeem their funds, they just need to pay a penalty of sorts for making the company find someone else to take out their position or, if no one can be found, using their own capital to cover the position of the exiting lender.
Sound like a reasonable business model?
Guess what? That's a bank, offering savings and money market accounts in the first instance and CD's in the second instance.
PART TWO:
People around here act as if the fractional reserve system is a crime, I just don't get it. Yes, it's true, if 100% of a banks depositors showed up at the door demanding to withdraw their funds in full on the same exact day, no modern bank would be able to accommodate that request. The fact is, though, that these events just don't happen anymore. And even if they did, such an event wouldn't necessarily mean the immediate insolvency of a bank - in their customer agreements the small print generally states that they can limit withdrawals in some instances - one instance would be depositors demanding all of their funds at once - so long as capital markets are functioning normally (like, we're not in a 2007/2008/2009 scenario), the bank might just need a couple of days to sell off assets that those deposits were invested in - loans, mortgages, credit card balances, those are all marketable securities. So, as long as every bank isn't trying to sell the same exact asset at the same exact time, a modern bank should be able to accommodate full customer withdrawals in a surprisingly short amount of time, without needing to declare insolvency.
Take JP Morgan for instance.
Balance sheet shows $1.287 trillion in "Other Current Liabilities"; those are customer deposits.
They also show $405 billion in long term debt; those are likely bonds and certificates of deposit - things that customers are either dissuaded from redeeming immediately (via a Certificate of Deposits' early withdrawal penalty), or simply CAN'T demand redemption on demand (in the case of corporate bonds).
To offset those two sets of liabilities, they have $604 billion in cash and equivalents, so they could accommodate almost 50% of their customer withdrawals from can on hand. I might take a day or two to shift the funds to the appropriate branches, though.
An on top of that, they have $1.5 trillion in long-term investments; these can be letters of credit, credit lines, mortgages, and the like; all of those are marketable securities which can be sold to other banks, to institutional investors (pensions, hedge funds), etc.
What am I saying?
Fractional reserve is not a crime or a fraud. If the business of banks was to hold customers cash and to keep it as cash, then what would be the point of banks? No, their business model has always been to take customer deposits and lend those monies to others of the banks customers. That's a completely valid business model, not fraudulent, not criminal or any of the other terms that get thrown about when they get whipped into a foaming at the mouth type frenzy.
Banks have found they can do business while only having a small fraction of their customers deposits on hand as actual cash, because even when many customers are requesting to withdraw funds, others are depositing new monies which offset those flows. No different than Coinbase determining that they can satisfy the day to day business demands of their customer while keeping 97% of their customer funds in cold storage.
PART THREE: Role of the Central Bank.
One of the most important things that the Central Bank does is establish reserve requirements that banks must abide by; they can do this because they have tremendous insight into the economy - insight that almost no one else has. They have this by being able to discuss, in complete confidentiality, current business environments of hundreds if not thousands of companies across a variety of industries. Information that wouldn't be disclosed to shareholders until it's time to make quarterly filings is disclosed to the Fed freely, so that it can have it's thumb on the pulse of the economy at all times. When they see that the economy is expanding, they lower reserve requirements, letting banks make more loans into the market and have less cash on hand, and when they see the economy is stagnant or contracting they raise reserve requirements, making banks remove loans from the market (that could be calling in debt, closing credit card accounts, lowering lines of credit, etc), in order to have greater amounts of cash on hand.
Sometimes they mess up. 2008 is a perfect example. But even in the face of that, it doesn't invalidate their role in the economy.
Circling back to the first point; fractional reserve banking is not an evil invention, despite all the hype to the contrary; and more importantly, as the bitcoin denominated economy grows, such services will not only spring up, but will likely be in demand; they will be able to provide services that just aren't viable in a peer-to-peer manner; why should hundreds of lenders do the same amount of due diligence on a potential borrower, when they could transfer funds that they're willing to lend to a separate company, where each of those other hundreds of potential lenders have also transferred funds, and rely on that one company to do the due diligence on their loans for them? The second scenario is far more efficient, and that, my fiends, is a bank.
And should these bitcoin denominated banks have a central bank?
To me, the answer would be "ideally", but only if that central bank was able to get equivalent or near-equivalent visibly into the economy as the true Central Banks do. There's no point in having an organization making rules the bitcoin denominated banks to abide by, if they aren't in a position of having more insight than each of those banks do. So initially, there's not really going to be a place for a bitcoin central bank in bitcoins earliest days. Until the bitcoin economy gets big enough, and more importantly, creditable enough that, upon formation of a "central bank" or its equivalent, which can actually glean more information about the economy than the bankers themselves, there's no point in having an outside organization make rules for those banks. But once bitcoin gets big enough that large companies are comfortable being candid with bitcoins' Central Bankers, those Central Bankers will be in a position, knowing more about the economy than the banks themselves, to be able to hand down general decrees to the bitcoin banks that had opted to operate under their umbrella.
That's about it. Sorry about the ramble.