The narrow job description and advertisements are applicable only to the green card where the total cost would probably reach over $8000, but again it is often shared by the employee and the employer. The job description is a few hours of work by an immigration paralegal or lawyer and newspaper ads should not increase the cost by much more.
I don't want the H1B system to be something it is not, the government has already muddled the intent of the system by allowing H1Bs to apply for greencards. In fact it is probably the only practical way of getting a green card as a skilled immigrant, but government chokes the supply of those at will, making it unpredictable. If the sole goal of the H1B is to relieve temporary shortage of labor, H1Bs should not be allowed to apply for green cards.
In the end, I think we have arrived at the same conclusions: that we should increase our immigration quota for skilled immigrants and establish a system independent of the H1B visa for it. I don't know how you intend to test for cultural fit, but I am open to listening to ideas there.
I was dreading putting the bonded labor comparison there, which is why I added the caveat. But a bonded laborer cannot leave their job until their debts are paid off, and an H1B with a greencard application pending cannot leave their job until they get a greencard (or risk losing whatever life/assets the have built up here). I see similarities there, although, obviously the two are not identical.
It is easy to say that immigrants should not expect to build a life and live here permanently, just because they can apply for a green card. However, the current approach to immigration causes uncertainty which hurts both the country (because very skilled immigrants, given a choice, will not accept the uncertainty) and immigrants themselves (for obvious reasons).