You are absolutely right that it would be madness to hire a programmer without having seen him code. The companies are well aware of it and they will surely ask you to write a lot of code during the interviews. But that comes at a much later stage, as it costs them much more money than checking on your resume whether you have a degree or not.
I'm not saying that I like the way things are, but that's just life. If that's of any consolation, in most other industries the requirement of having a degree is much more strict than in case of Software Engineers.
No one but a programmer is qualified to evaluate another programmer, HR and recruiters have no place here and they'll just waste time and effort pretending to be useful, they aren't.
By doing the test you've already filtered out the wanna be's so you don't need phone interviews, the submitted samples should go strait to a qualified programmer, it'll take him only a few minutes to toss out any bad submissions and he'll quickly know who's worth actually interviewing and who's not.
Code is the only resume a good programmer needs.
Can you imagine a test for the ability to work on larger programs?
Again, the test is just to weed out the bad candidates and allow you to only interview worthy people, everything else you'll find out in the actual interview. Interviews will be rare, few people will submit code you'll find acceptable at all. The vast majority of people applying for programming jobs, can't program, it's sad, but true.
On reflection, I think you may be right that good code is good code. Size of the system matters most when the code is bad.
Nothing makes for a better interview than doing a code review and critique of the candidates own code. Something simple but telling like write a little slot machine that lets you bet money, spin the wheel, and win or lose with the game ending when you run out of money.
Anyone who refuses such a test isn't a programmer you want anyway, and you can tell an enormous amount about a potential candidate by simply looking at the quality of the code submitted. Don't bother interviewing anyone who's code you can't stand. You can see everything you need to know about their skills in that code.
You wouldn't hire a graphic designer who refused to submit samples of his designs, nor should you hire a programmer who refuses to submit samples of his code. Merely by having such a test, you'll weed out all the fakers because they won't bother with it, they'll submit their fake resumes elsewhere.