No, the NIST curves simply can't use random strings as points at all: randomly chosen vectors can be points that aren't valid on the intended curve, but rather on curve of much smaller order; repeatedly submitting insecure curve points can allow attackers to recover private keys via CRT. It's a pretty awesome attack and not hard to implement in a test environment.

The unbiased randomness issue is with the ECDSA construction, which requires a totally, uniformly random k value for every signature. Bernstein's Ed25519 construction is deterministic, and doesn't have this problem.