Under what circumstances that Liquid-Liquid extraction would be good to use?
The above reasons to use LLE are focused on the compound thermal similarities and instabilities.

But it does not take into the account the amount of compounds. While distillation separation can be great for major components, it may fail for minor one, and definitely for trace ones.

The LLE is frequently used for separation and content enrichment of trace amounts, in wide range of grams to nanograms per litre.

Using the approach of chemistry of complexes, it may also lead to great separation of non-volatile compounds/ions/elements.

LLE is e.g used for mutual separation of  nuclear fission products and not spent fuel by the PUREX method.