Drop both the number and the hyphen and the thing should be self-explanatory. Either way, it’s explained well at Englishclub.com/grammar/nouns-adjective.htm
The rule is simple, although the explanation might not be. As illustrated in your list, the rule is that nouns used as adjectives are always singular. Compound adjectives are merely a class of nouns used as adjectives so they, too, are always singular - except in irrelevant special cases - and normally, if not always, hyphenated. Grammar-monster.com/lessons/hyphens_in_compound_adjectives.htm reminds me that UK readers like me will expect hyphens by default; US readers might be more lenient, expecting hyphens only to eliminate ambiguity. That could be beside the point.
No-one would normally speak of item plates, person tables or even man races, though grammatically those would be no different from name or soup plates, dinner or work tables, car or horse races.
“Two-item” corresponds to “soup”; “three-person” to “dinner” - though “dining” would be more usual; “two-man” to “foot”.
There will never be a “two-soup plate” nor a “three-dinner table” or a “two-foot race”, even as compared to a “three-legged race”.
The first noun is used adjectivally to qualify the second, the subject. The number might change the size but never the intrinsic nature of the subject. A “twenty-item plate” remains in essence a plate for items, even if it must be larger. The same is true of tables and races.
Even plural things such as taxi ranks or car parks, race circuits or dog tracks or passenger ships take singular adjectival nouns
It makes no difference whether the plate is designed to or merely happens to hold two items; whether the table is meant to or only happens to seat three persons; whether the race was a specific challenge between two individuals or happens to involve only two runners worth the name.