Also, you create a fake shortage by driving wages down: "we have so many positions that we cannot fill (at the price that we are willing to pay)"

Yes if it's real. No if it's fake.

This still ignores the question of if entry limits for jobs are set reasonably or not though, you can create a real shortage by limiting yourself needlessly. At which point those who think that it is needless will call it "fake", so which word you use also depends on opinion and interpretation of data :-)