Is it safe not to record private keys for paper wallets? That may sound like a silly question, but if I have created my keys using a passphrase in “brain wallet” at bitaddress.org, why do I need to record the private key, if I can recreate it at any time by using my downloaded version of bitaddress.org?

To be sure that I can do that safely, I need to know whether the algorithm to generate a private key from a passphrase with a downloaded copy of bitaddress.org is deterministic.
To answer your question: bitaddress.org uses the SHA-256 hash of the password you provided as the private key. Therefore, you can always derive that private key from your password. If you can write a few lines of code, you can test this yourself (I just did).

However, if you generated your passphrase using dice then you are unlikely to memorize it. Therefore, it's not really a brain wallet. The SHA-256 step is really unnecessary, you might as well use the numbers from the dice rolls as a private key (you'd have to convert it to binary, encode it as 32 bytes and then convert it into the wallet import format).

The only reason to do what you just did is if you don't trust the random number generator in your browser. Now that you have a private key, you might as well write that down and forget the input to the SHA-256 hash.