Chinese can be a singular or plural noun.

I assume that the possessive of the singular is Chinese's.

I saw a Mexican and a Chinese. The Chinese's jacket was green.

What's the possessive of Chinese plural?

Italians' favorite food is pizza but _____ is chop suey.

It cannot be Chineses'. That would be double plural. Could it be Chinese'?

Consider this.
Your examples sound like sentences that, in my experience, only a Chinese person would say.
Natural English would be
eg I saw a Mexican, and a Chinese person. The Chinese person's jacket was green.
eg Chinese people's favourite food is chop suey.

I don't know why we don't often use the term the way you suggest . . . but we don't.

We do sometimes speak of all Chinese people as 'the Chinese'. But I think we avoid the possessive form, possibly because we find it a bit confusing, too.
Thus, we'd typicall say

eg The Chinese are usually very polite.
eg I am impressed by the politeness of the Chinese.

Logic doesn't apply to the genitive in English. Certain things are frequently said and therefore they sound natural. There are nouns that are never used in the genitive. For example, you never hear people say: This is the French's favourite drink. Nouns ending in ese don't usually take a genitive s. If they did, English genitive rules would give us the following:

Singular, one Chinese: The Chinese's jacket was green.
Plural, more than one Chinese: The Chinese's jackets were green.

There would be no difference. The singular genitive is formed by adding an apostrophe and an s: Chinese's. If a plural doesn't end in an s, in other words the plural is irregular, the plural genitive is formed in the same way, by adding an apostrophe and an s: children's books, men's shoes, and Chinese's jackets. This isn't used, though.

You had better say: the jackets of the Chinese. English grammatical usage is often very illogical and rules are often violated and the resultant language may sound very natural and idiomatic. That's one of the reasons English is so difficult and fascinating.

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Conversationally, you are 100% correct. But what about the elevated prose style, where the use of circumlocutions to avoid grammatical difficulties is often obvious?
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Cool BreezeSingular, one Chinese: The Chinese's jacket was green.
Plural, more than one Chinese: The Chinese's jackets were green.
Yes, I can see you're right.