Even if they did check things, I've seen write ups on how to abuse the system, like writing job descriptions for senior level positions but listing the job as a more junior level position.
So, they list the job as a "Junior business analyst" or even "business analyst" and then when Uncle Sam comes knocking, they can say "we're paying market rates, here's the BLS data for the position" even though if you looked deeper, the employee is expected to do much more.
I understand this bit, especially in a litigious society.
On the other hand though, you cannot say that there is a "talent shortage" if you are unwilling to consider (and ask about) this type of work. I know I have a little bit of a speech impediment/stutter and it gets worse when I am in an uncomfortable/unknown/nervous situation. But, it should have no issue programming and working with others. An interview like the OP suggests would quickly defuse the situation, and show that I am competent. In other words, it would level the playing field.
Oh, I have heard the horror stories about people claiming to have senior level skills but perform at a junior level, or below.
The feedback I tend to get is that I am generally not as experienced as some of the other people they generally get, but I generally get good feedback - I don't throw a hissy fit if I don't know something, and can explain things well enough that if I don't get the syntax/algorithmn correct exactly correct.
While I don't doubt there's a lot of people that are on the low end of the curve, I think part of the problem is are these:
(a) interviews. I know I have bombed interviews where they ask about some problem or data structure that you haven't touched in years. I could tell you the theory behind recursion, and figure out an inelegant brute force solution on the spot - if I had access to an editor/compiler/debugger I could eventually figure out the more elegant solution.
(b) I think people become API monkeys because in many cases, why invent the wheel. Unless you're building a Kernel or some other special case application from the ground up, you're going to want to use what is most efficient, most readable, least chance of bugs, etc.
(c) I think people recruit the wrong way. They usually throw up a job ad somewhere and let everything roll in. Many people who you want to hire aren't necessarily on those sites. Or you get some clueless 3rd party recruiter. Many times the people you want are members of user groups or other prof. organizations. The people there are generally higher quality, more willing to learn, etc. And you can get to know them in a lower pressure environment.
Is every taxi driver and burger flipper without any relevant education/experience applying? I would agree that these people are unqualified.
However, if you're getting people who have a relevant education or maybe not the exact experience you're looking for, you'll have a harder time convincing people that they are unqualified. That's more of a gray area. For example, someone who has a computer science degree and does C# applies for a Java job. Their Java may be rusty but the concepts of programming, algorithms, OOP, etc. don't really change. They may appear to be "unqualified" if they are rusty but should be able to adapt.
I think there are people out there that know nothing, but it seems to me that there are problems on both sides of the equation. It seemed to me 30-40 years ago, corps were willing to work with you even if you weren't a so called rock star.
As I have commented here before, the system is also broken and it is perpetuated by everyone. For example, any Joe can apply to 20 jobs a day because corporations solicit that and make it "easy" thanks to the Internet. Of course, they write "complex" software to filter which may not work well, so people feel a need to game the system/cheat to actually talk to a decision maker. Added to the problem is HR/3rd party recruiters who are not technical have inserted themself into the process.
Personally, I'd rather hire someone that just told me "you know, I don't have every single skill listed on there but I have a lot of them and I've proven that I can learn quickly".
The problem is that most software will filter these people out. Or HR/Recruiter is too stupid to know the difference between good and bad. Therefore, people feel the need to game the system to even talk to a human to actually gauge what you really need. In some ways, I don't blame them.
And this is the crux of the matter, wish I had mod points.
I think part of the problem is that we don't teach/train people well enough - that's why it appears people aren't qualified.
And on the other hand, we want someone with 90th percentile skill but will only pay them in the 50th percentile. That, coupled with the point above makes everyone scream "shortage"
Of course training needs to be improved, or at least there is some room for improvement.
My issue is that corps talk a big game - there there is a shortage of qualified candidates. What there is a shortage of is good training, planning, career paths and adequate salary. If there was really a shortage, we'd see changes in these areas.