I can't comment on bitcharity.org, but one thing you could do is inform the charity of any donations to those addresses (and the charity themselves could monitor transactions to those addresses).
I disagree that tax-deductible anonymous donations are quite common. Sure anyone can accept bitcoins, but to meet the IRS rules, either the deductor or the charity have to have records. As far as a charity handing out blank tax deductible donation receipts, that wont work with the IRS where you just fill in the amount and they accept that position. I'll admit not all businesses know all regulations, and not all filings and records are in proper order. But if you look at the rules set out, they dont entitle anonymous tax deductible donations.
Anonymous donations are very common. You are correct that either the person claiming a deduction or or the charity needs to have records. I think you're confusing anonymity and record keeping. As far as the IRS is concerned, they only care that you can prove to them that you made the donation (the proof can be in many forms and is only actually needed should you get audited). The charity does not have to know who made the donation. In fact, my previous employer allowed donations through payroll deduction and you could select an option to remain anonymous. Obviously, the employer would know who made the donation, but the charity would not. And those donations are definitely tax deductible.
I don't believe that that is correct. I've worked the charity side of audits a number of times and the auditors always insisted that the charity have a record of who made the donation for their tax purposes. I'm not saying this is definitive but it's always been the case with charities that I've worked with. I'd recommend that you talk with a reputable non-profit audit/tax firm to be clear on the rules. It's a generous offer, but please make sure you have your i's dotted and t's crossed.