Detection of weak keys for AES-GCM
However, none of these suggestions have been accepted in the NIST standard.


The suggestions are to use a different group. Those are not changes that could be done to the standard without complete loss of compatibility. They are things to consider when designing and choosing new algorithms, however.


  Or is the probability to get a weak key (assumed that one is using secure random number generators) still so so small that the existence of weak keys is negligible?


Yes. From Saarinen's risk analysis:


  The probability of randomly hitting an exploitable weak key with a real-world AES-GCM cryptographic protocol such as SSH [4], IPSec [6] or TLS [12] is vanishingly small.


No protocol that I know of uses GCM with a 64-bit cipher (like 3DES or Blowfish), which Saarinen calls out as risky and recommends against.