"Pseudo random generators are quick but predictable": well, that's true only in an ethereal, mathematical sense. In practice, if your PRNG is cryptographically secure and seeded with some initial randomness of large enough entropy (128 bits are enough), then it won't be predictable. However, if a PRNG is involved, this is not true "One-Time Pad" but a mere "stream cipher".
OTP is a nice theoretical toy but it has no practical advantage over a properly used stream cipher. For some reason, a lot of people have fantasies of OTP being the ultimate answer for the Real World, and this leads to many misguided designs.
The main practical issue of OTP is that the key must be as large as the message, which is inconvenient: usually, we use cryptography to reduce problems: we apply encryption to reduce the problem of ensuring the confidentiality of a 10 GB file to the problem of ensuring the confidentiality of a 16-byte key, which can be managed much more easily (it can even be remembered and typed in by a human). If the key is as large as the data, then no reduction has occurred. At best, OTP can bring a time-based advantage: exchange the key in advance, when conditions allow for that, allowing for secure transmission later on. That's how the red phone worked: tapes full of random bits exchanged on a weekly basis (by special planes), so that immediate secure communication can be done at any time with no latency.
Of course, OTP relies on the key stream to be secret. There is no point in using public data as key stream. The whole security of OTP relies on secrecy of the key.
There is, however, a generic key exchange protocol which works with a public source of data similar to your "number stations".
It works like this:
- Alice and Bob listen to the number stations, and write down, at random intervals, some of the numbers and the "index" of these numbers. For instance, Alice remembers "171th number was 40; 236th number was 91; 406th number was 38;..." while Bob does the same with some other randomly chosen numbers in the public sequence.
- At the end of the day, Alice sends to Bob the list of indices that she remembered ("I have numbers 171, 236, 406...") and Bob responds with his own list of indices. With high probability, they have a few indices in common, so they will use the corresponding numbers as shared key.
An eavesdropper, however, would have to remember all, or at least most of the numbers, in order to also remember the same numbers. If the "number station" is sufficiently high-speed (say, it outputs 1 terabyte worth of data per second, and Alice and Bob gather randomly picked numbers during one whole day), then the storage requirement for the attacker can become prohibitive.
The Cachin-Maurer protocol is a fine example of a theoretically secure but grossly impractical cryptographic algorithm. Just like OTP.