In Europe a 2013 law is now in effect which basically states no more bank bail-outs, from now on it will be bail-ins. In other words if a bank fails it will use its customer's deposits to cover its debt.
The US, EU and many other countries have deopsit insurance* schemes that limit risk to depositors. You may not get all your money back if you have large deposits, but for example in the EU you're generaly guaranteed to get at least the first 50,000 Euros.
There is no way not to be the potential victim of this, because bank customers do not have enough information to determine whether a bank is sound or not. Worse, soundness is dependent on what everyone else is doing; a bank can be fine one day then destroyed due to a bank run. Risk is hard to deaggregate in the mortgage business because house prices all move together.
You're onto something with "profits of risky loans are privatised while the losses are socialised", but bitcoin really isn't a good solution to that.
any benefit is on average far, far outweighed by the costs
A cost-benefit analysis with no numbers! Convincing!
Bullshit. Customer reviews, third party audits, history and reputation, secondary signals (quality of website, customer support, advertising, quality of investors). The list goes on  Your only limit is your imagination and your confidence in the intelligence of your species my friend. You've been tricked by the miserable current state of banking (a product of socialist intervention) into thinking that this is the way banks are and must be. You'd see how quickly things will change when a) banks live and die by their reputation b) anyone can build a bank from anywhere to service the world (enabled by Bitcoin.)
bitcoin really isn't a good solution to that.
Why not? Bitcoin supersedes political violence. No government can order more of it to be created at gunpoint. Therefore there is no way to enact "backdoor" taxes (inflation) to support inherently-broken schemes such as socialised deposit insurance. The only way is frontdoor taxation, which people are far better at measuring and disliking. The success of Bitcoin would cut off the last life support tube for dysfunctional government
 Yes, I know you'll have point by point reasons why each one of those techniques is somehow uniquely impossible to work with the 100% success rate uniquely required by this and all other regulated industries (health, inter-city transport et al), which for some reason work just fine in the industries which escape regulation (online commerce for instance) which for some reason are much more forgiving.
What if your bank is not a member of FDIC ?
It's sort of what the European Union has with those bail ins, deposits are guaranteed up to 100k but this comes from inside the bank, first from shareholders and creditors then larger accounts. Makes sense that the public should not have to pay for the failure of a private banking entity.
Then don't do business with that bank. The vast majority of banks and credit unions in the US offer insured accounts. If yours doesn't you should move your money elsewhere.
The FDIC insurance is $250k per insured-account-type per institution. That is, if I have a checking and a retirement account with one bank I'm insured for a total of up to $500k. Anything exceeding that needs to be in another account type or another bank to be insured. Most investment accounts aren't insured, but some brokerage accounts are (you'll get that info when you talk to your bank) but the investments themselves (naturally) are not, just the value of any non-invested cash.
The cost (in the US) is put upon the banks to pay for this insurance. It's just a federally created insurance agency. Barring a major collapse across many (most?) banks, it should have enough stored up over the years to cover the cost without requiring financing from the federal government. In theory, you can have private insurance (I believe these do exist) for your bank, but I'm not sure how common this is in the US.
Like most insurance schemes, the more participants the more affordable it is. It'd be a tough market for someone else to break into.
We despise existing solution thinking that we could and should disrupt everything. Tear down everything and build everything from scratch; Our way or the high way attitude.
Bitcoin definitely have some innovations here and regardless of what happen in the future, it's impact will be felt. It was a great experiment, probably the greatest social experiment of the 21st century. The best is that it's self contained and sandboxed. Allowing it too roam freely in unchartered water.
But the financial industry is not as un-innovative as we hope they are. To innovate means you have to experiments, to experiment mean there will be failures. Over the years we seen financial industry fail and fail again. In fact, the incentive to "innovate" is so high, you constantly see the cat and mouse game between the regulators and the industry players, and the revolving doors of the prison cells never really stop.
While the greed and failure of the financial industry has irked us and certainly bred a level of mistrust. Trust in the financial system is at the highest level in human history. You have to understand that they are no stranger to mistrust and throughout human history it is a recurring problem they have to solve over and over again.
The current floating currency and central banks structure which provides deposit insurance among many other things are the result of many iterations over thousands of years. This system run the global economy, it scale is unimaginable large in human scale. There's a lot to appreciates.
Sure there's room for improvements. Many people treats Bitcoin like it must replace the financial system but the problems it facing now is exactly what the financial system had solved. What I'm pretty sure is that Bitcoin's will definitely had an impact on the next iteration of the ever evolving financial system.
The Bitcoin community I have seen fails to recognized the important of trust in finance. Not just trust between computers, but trust between humans. Like the 90's and 00's software industry fails to recognize the important of simplicity and good user experience. The next Steve Jobs of the cryptocurrency is in my believe one who understand these ideas to heart.
There's a lot to celebrate about Bitcoin and what we had done, as I always like take a contrary position. Bitcoin is not a replacement of the existing financial system, it is, or will be if it haven't, part of the system. Thus, the rise and fall of its price is part and parcel. It's kinda exciting to see this great experiment in the running, although I'll dread experiencing it first hand.
As Bruce Lee once said, "In great attempts it is glorious even to fail." I wouldn't consider this a failure but it's definitely glorious.