Should supervisors let student authors make mistakes? And should reviewers care?

Here's an issue which I've encountered occasionally as a referee over the years (though not recently, and not as a handling editor as far as I can recall). It concerns manuscripts for which a student is the lead author, and their supervisor is a co-author. Once in a while I find that such a manuscript contains one or more serious mistakes, such as confusion about basic concepts, an experimental design that completely confounds key factors, failure to measure important response variables that obviously should've and easily could've been measured, or serious statistical errors such as analyzing a nested design as if it were a factorial design. The nature of the errors is such that I would not expect to encounter them in papers lead-authored by the supervisor.

So my assumption (and I emphasize that it is an assumption) is that one of two things is going on.* Either the supervisor didn't really read the paper carefully before it was submitted, and so wasn't fully aware of the mistakes or of their seriousness. Or else the supervisor was fully aware of the mistakes, but decided that "it's the student's paper, let him make his own mistakes". And of course, these possibilities aren't mutually exclusive, since a supervisor who gives his students a lot of freedom and lets them make their own mistakes is the sort of supervisor who might let students submit an ms without first reading it carefully.

My question to you is: are you bothered by this? Because I am, but I'm not sure if that's just me. I'm bothered for several reasons. First, either possibility I've described would seem to be a violation of the published rules of most journals, which require that all authors take responsibility for everything in the manuscript. Second, even if those journal rules didn't exist, wouldn't you still want to make sure that any science with your name on it was correct? Third, I'm most bothered by the apparent willingness of some supervisors to effectively force reviewers to do the training that the supervisors ought to be doing.

Note that the situation is totally different if the supervisor isn't a co-author. As a reviewer, I'm not the least bit bothered if I'm reviewing a manuscript sole-authored by a student and find serious mistakes that a more experienced author probably wouldn't make. Note also that I'm all in favor of allowing students a lot of freedom, including the freedom to make mistakes. But that freedom does not extend to the freedom to make serious, clear-cut mistakes with my name on them.

But then again, maybe I shouldn't be bothered by this. One could take the view that it's the job of reviewers to identify mistakes, no matter what the source of those mistakes. Further, even very experienced people do sometimes make serious mistakes (like believing in zombie ideas!), so maybe my annoyance here is based on the false premise that there are some mistakes that should just never happen in any paper with an experienced co-author.

What do you think? Should supervisor co-authors let student lead authors make serious mistakes? Should reviewers care if they do? Or is this whole post just based on a false premise?

*Actually, I suppose there are at least two other possibilities: the supervisor is aware of the mistakes and their seriousness, but either hopes the reviewers won't notice or care, or else hopes to be given the opportunity to fix the mistakes in a revision. But I ignore these possibilities, because considering them is too depressing.

p.s. to my own students: this post was not inspired by you!