I am sorry that I have not done much work on frailty terms in Cox models. I was, on the other hand, not that informed of the rationale underlying the claimed deprecation of the use of log-likelihood in the presence of frailty terms. I think attaching links of the information you cited (including the claimed invalidity of AIC in the presence of fraility terms) is of use for other users in this forum to look further into the details if they feel that are not fully informed of the rationales after reading your post.
I agree with @Ksharp on the validity of the likelihood ratio test in Cox models in the general setting. But caution should be taken in your analytic project, as you have pointed out the possible invalidity of the use of likelihood for Cox models in the presence of frailty terms. Likelihood ratio tests are based on the comparison of log-likelihoods. If the use of likelihood is deprecated, then perhaps the use of likelihood ratio tests are deprecated as well. I used the word "perhaps" because I have not searched for evidence personally, so you are suggested to search the web and find for evidence yourselves.
If both likelihoods and AICs are not valid in this case, then there is another graphical tool for the assessment of goodness-of-fit of Cox models. That involves the plotting the logarithm of the Cox–Snell residuals against their log-log transformed probabilities. See page 215 of A Handbook of Statistical Graphics Using SAS ODS for a detailed code of how to do this in SAS.
