If you use k-means to cluster your data into 100 clusters, it will do so, irrespective of whether it is meaningful to do so. Perfectly objective, but what does that objectivity buy you? If your pet theory is that there are 100 groups, you'll be actually less likely to get results that disagree with that than if you ask an LLM how many groups there are.

But the real question is not whether you agree with the results, but whether they're useful. If you apply an objective method to data it is unsuitable for, it's garbage in, objective garbage out. Whether the method is suitable or not is not always something you can decide a priori, then you need to check.

And if trying it out shows that LLM-provided clusters are more useful than other methods, you should swallow your pride and accept that, even if you disagree on philosophical grounds. (Or it might show that the LLM has no idea what it's doing! Then you can feel good about yourself.)