What are the risks of using CTR mode with 64 bit blocks?
I believe that he was referring to the misconception that the birthday problem that arises in encryption is only when you use the same counter twice. If a random IV is used, then such a counter repeats at $2^{32}$ blocks with high probability (and if you want a $2^{-32}$ safety margin then you can only encrypt $2^{16}$ blocks). However, if distinct counters are guaranteed to be used, then you never get a collision on the input to the block cipher and so you can encrypt more. I assume that the NSA claim was with reference to this version of counter mode where distinct counters are guaranteed (something that's easy to do if you have state).

Anyway, this is a misconception for the exact reason that you point out. The problem is that people think that this leaks only a very little bit of information, unlike in CBC mode where a collision in the ciphertext immediately reveals the XOR of two plaintext blocks. Recently, at EUROCRYPT 2018, a paper was presented showing how to obtain actual plaintext in a number of realistic scenarios when counter mode is used beyond the birthday bound, even with unique counters. The paper is called The Missing Difference Problem, and its Applications to Counter Mode Encryption. I recommend reading it.