For liaison, it's helpful if you remember that the writing reflects the spoken language, and not the other way around.
So, hundreds of years ago, when the z in chez was pronounced in all contexts, one would say che/z/ lui, che/z/ elle, etc, che/z/ Louis, etc, all with z pronounced. When final consonants disappeared, they were retained before vowels (because the enchainement process meant the "z" before elle was pronounced as the beginning of a syllable rather than the end of the preceding one).
So if a phrase is relatively "new," we expect it not to take liaison (or only to take liaison in educated contexts), whereas if it's old enough to predate the sound change, we expect it to take liaison. In this case, "chez elle" and "chez eux" are both standard prepositional phrases that one hears all the time; this reinforces that the liaison should take place. By contrast, "Chez Alice" refers to where a specific person Alice lives. Unless Alice was alive when the sound change happened, we don't expect a liaison.
This is also why lots of fixed phrases like "de temps_en temps" take liaison (the "s" was pronounced before "en" when this phrase acquired its fixed meaning), as do standard grammatical constructions "vous_atterrissez." Grammaticalized liaison takes place even before "new" verbs, presumably because the liaison rule is so uniform and internalized in these contexts: "vous_alunissez."