I know it's frustrating, but there is something you can do about it, if you're willing to be a little flexible on dates.
From the NPS Grand Canyon website:
You can stop by the Backcountry Information Center at any time during open hours and request a waitlist number. This number is valid for the following morning and will be used to determine priority of service. At 8:00 a.m. Backcountry Information Center staff will call waitlist numbers. When your turn comes you can request a permit, exchange your number for a new waitlist number good for the following day, or simply ask questions. You may participate in the waitlist for as many consecutive days as is convenient.
My wife and I applied for corridor campgrounds over Spring Break, but (suprise! surprise!) we didn't get them. Since we had a week at the canyon, we decided to chance it using the waitlist system. We left on Friday as soon as we got off work, picked up my brother-in-law at the airport along the way, and drove through the night to get there Saturday a little after noon.
When we walked in to permit office, there were several sites still open that whole week, and we got incredibly lucky--two nights at Bright Angel for one trip and one night at Granite Rapids for another. Now, a lot of that had to do with luck, but we also beat the rush for the Spring Break crowd so there was much more open for the walk-in permits.
I also found the rangers to be far more helpful in person. The people we talked to told us what was still open, talked us through the options, and were really easy to work with. If I remember, the BIC does not take phone calls.
If I were you, I would go anyway (assuming you have a week or so to be there to give you time to work the waitlist system), study up on all the places you'd like to try, not just the corridor trails, and more than likely something will open up soon for a 2-3 day trip (maybe more).