If the client paied for prepress preparation the printer should take responsibility. Otherwise it is printers job.
What would the case be if the client was going for a special effect, and wanted the overprint, and the printer noticed the overprint, thought it was an error, and removed the overprint, and the project got printed wrong?
Surely you can't be serious (some Airplane humor). Overprinting elements occur on Flexo jobs all the time. Intentional (to create an extra tone from 2 spots) or not (element is too small to trap with Flexo standards). The results can drastic as the inks we use are pretty translucent as well.
Flexographic printing is a different bird though.
Tough one. I guess your stance depends on how important the customer is. Our customers (and sales) rely on prep to catch missed elements (classic one: white text set to overprint in Illustrator) but we proof a PDF showing traps and overprints so the final approval belongs to the customer (there is no "ok with changes" allowed).
The customer viewed their PDF without overprint, the default in reader. The supplied file was a PDF. The job went to press without customer approval, as the vast majority of our jobs do.
This is where you went wrong. If the customer had approved it then all reasonable checks would have been fulfilled. It doesn't always prevent everything but a <$30 proof is almost always cheaper than a reprint.
I agree that most companies would end up eating the job, as there is a gray area when it comes to whose fault it actually is. Needless to say, both sides have some sort of liability, and obviously neither side will want to shoulder the blame.
When situations like that arise, I find that it's usually better to take the lumps instead of losing a potential customer.