You're assuming the print company is the one making the mistakes.
If the printer is your contractor, you are holding the whole bag. It's up to you to deliver what the client wants, on time. You need to develop relationships with various printers to make sure that happens. Always include the cost of a proof in your contract. If the client doesn't want to pay for (or doesn't have time for) a proof, have them sign a waiver that you are not responsible for the outcome. Always include a schedule in your contract so the client knows when they have to meet deliverable deadlines and how long they have to approve proofs. If the job has a very quick turnaround, they should even be on-call.

If the client is doing the printing, state in your contract that you must have contact with the printer in order to prepare files to that printer's specifications. If no contact, at least an up to date spec sheet from the printer. If the client refuses to supply, whip out that waiver.

Not all errors in printing happen because of the printer. It's entirely possible files are set up incorrectly, there is no time allotted to do the job properly, no proof is requested, and the client is sometimes not available to answer questions or provide links/fonts/approvals in a timely manner. Any glitch in the process on your part, moves your job aside and out of the queue. It pays to know your printer and work with him in scheduling and in preflight. Don't settle for cheapest.
You can have Cheap, Fast, Quality Work. Pick any two.

I wasn't suggesting that it's the printers fault every time. But sometimes.

Like me, printers are people too and are prone to mistakes from time to time. Same thing goes for unpreventable problems (i.e. natural disasters, power outages, fires, theft, etc).

As the designer, what relief can (or should) I offer to a client, when the normally reliable printing company fails to do their job?

What if the printing company fails because, oh, say, a plane crashed into their building? Then what?

Would I still be responsible if the delay is not my fault, nor the printer's fault? The client has no business relationship with the printers, so by default, the blame is mine regardless. Do you see where I'm going with this? How do I get around that?

In all of those situations, it isn't an issue between your client and you, it is an issue between you and your printer. Make sure you only use printers who will bend over backwards to keep your business if something goes wrong (things do go wrong, even with the best printers). If they send you printing that is clearly not up to standard, ask for a refund or reprint and find someone else for your next job.

About half of the profit my business makes comes from outsourcing offset and digital printing, so it is definately worth doing - but make sure you add appropriate margins to cover your time and overheads - don't just break even - you are doing your client a massive favor by saving them having to find a printer and go through the whole process of getting their work done, and you have knowledge about the whole process that is a valuable asset to you, and worth charging for.

I don't know what to tell you as far as legalities.
Me, I always keep the lines of communication open with my sub-vendors. Printers have my home phone number (clients don't).

I also tend to have a plan B. It happens sometimes. Maybe one printer can't get a certain media in stock in time for any number of shipping reasons but someone else has it on hand, or a print machine goes toes up on the nightshift but another vendor over there has a similar machine. A lot of stuff I do has a drop dead deadline. Most of my printers have outsources of their own and handle a lot of this kind of thing, sometimes with you never knowing. But acts of Gawd... tough call.

One thing you might want to check into is insurance coverage. Say you drop that $10k order in a really big puddle or it flies off the roof of your car cuz it should have been in the trunk. There is liability insurance for that kind of thing. The lower the deductible, the higher the cost of the insurance. I don't handle that part of the business so I don't know what to tell you but a good accountant or maybe even the business office of your bank could help you out.

The other thing too is to protect your assets by becoming a company. An LLC or an Inc. or whatever is good in your state. You don't want your client to be able to sue you personally against your private assets.