However, you never see it used with other words ending in an s,
The issue arises, in general, with singular nouns that happen to end in s. There's no debate about "The boys' coats" because "boys" is unambiguously plural, nor about "David's coat" because David is singular and doesn't end in S. The issue arises around James because it's singular, therefore would normally take 's, but already has an s.
Pedant extra. You're often told that apostrophes do two different jobs: possession and elision (missing letters). In fact they only really do one: elision.
The apostrophe arrives in English for possession quite late (18th century?) because pedants of the time noticed that English used to inflect nouns for possession, and the apostrophe marked where an e had stopped being used.
If you look at the opening of R3, where a modern edition would say "He capers nimbly in a lady's chamber" first folio has "He capers nimbly in a Ladies Chamber", and where a modern edition has "Of Edward's heirs the murderer shall be", first folio has "Edwards heyres the murtherer shall be." Two different ways of forming possession, but no apostrophe. It's only used in first folio for things like know'st, from knowest, and lov'd, from loved - where the e would have been sounded at the time. These are mostly for metrical reasons.
To see where the C18 pedants were coming from, the first folio has "For Gods sake hence, and trouble vs not," which is arguably a contraction of "For Goddes sake"; after all, Chaucer has "For Goddes sake, to letten of his wille". So the C18 pedants formed a load of rules about apostrophes based on where, they thought, middle English would have had an e. They might even be right.
So if you think Chaucer would have had Jameses coat, it's James's coat. If you think Chaucer would have had James Coat, it's James coat. For it to be James' coat, you'd need Chaucer to have written Jamese coat. I don't know enough Chaucer to comment.