As far as I understand it, the very same problems would arise, if a hardcoded key would be used with any of the other keyed PRNG standards (like HMAC_DRBG or CTR_DRBG). What am I missing?
The issue is that, for X9.31, the key really is a long term key; it is never updated as a part of the RNG process.
That is, the state of X9.31 consists of a 'current state' (64 or 128 bits), and a 'key'; when you generate the next batch of outputs, you update the 'current state', but not the 'key'. Hence, if you start with a fixed key, you'll end up using that for the life of the PRNG.
So, if we assume that the X9.31 has been used for a while (and so the attacker doesn't know the exact state; one of its input is the 'current time', that does act as a weak entropy source). However, if X9.31 starts with a fixed key, that key is still in effect; if the attacker sees one output and has a guess of the current time, he can then predict future outputs (and previous as well, the state update function is invertible if you know the key).
In contrast, for the HMAC_DRBG or the CTR_DRBG generates, when they generate a batch of output, they update the 'current state', but they also update the 'current key'. So, if the PRNG is used for a while (and so the attacker doesn't know the state of the PRNG), the attacker won't know anything about the HMAC_DRBG or CTR_DRBG key, hence the attacker has no advantage in predicting future (or previous) outputs.
Of course, if the PRNG starts off with no entropy (fixed key and state), and doesn't get enough entropy before the attacker observes the output, the attacker can easily recover the state (and predict outputs); this is true for any PRNG. What is special about X9.31 with a fixed key is that this also holds even if the state has been given enough entropy.