Stative verbs in Chinese: only for adjectives?
As a native speaker, I have noticed such a phenomenon for many years, but I have never thought of the logic or reasons behind it. Perhaps we should not (or even could not) go too far on this topic.

This topic is academic, I believe. You would find many papers on this topic when you search on the internet (I use the key word: "汉语 形容词 谓语"). After reading some papers and thinking of how I would say in different cases, I think we'd better say "some adjectives could be used as the predicate in Chinese" and there are some fixed patterns where an adjective is used in this case. I would like to elaborate on this topic from several points.

1. Types of the Adjective
As Claw pointed out, there are two types of adjectives in Chinese (though, in academic world, there are some disputes on this. Someone holds that the adjectives (the term used now) in Chinese should be classified into two different parts of speech): "attributive" and "predicative". The difference between the two types is: "Can this adjective be modified with 很 (very)?"

2. Patterns for Attributive Adjectives
You could use the construction "[subject]是[attributive adj]的". In English, the verb "be (是 in Chinese)" is indispensable and it's called "Link Verb". In this case, the use of the adjective is similar with that in English.

3. Patterns for Predicative Adjectives
The problem becomes a little complicated now. I can come up with some patterns.

1. degree adverb (很,非常,十分,有些,有点,不) + adj. 
This is the pattern your examples use. In this case, I agree that the meaning of "很" weakens a lot. It seems that "很" works like a "marker". Of course, you may use "非常" or "十分" to express "very" if you really want to.

In general, it sounds very wierd "我好" ("你好" is another story, since it has been a usage or one interjection word, I think), "他高" in Chinese, but I think in some cases they are acceptable.
a. In a question and in the reply to that question.


  Q: 他帅吗？ Is he handsome?
  A: 他帅。He is handsome


b. Used in two clauses to express parallel, comparsion or condition.


  他胖，我不胖。 He is fat, (while) I am not fat.
  你高兴，我就高兴。 (If) you are happy, I am happy.
  这篇小说精彩，那篇也精彩。 This novel is excellent and that one is excellent too.


c. 又+adj+又+adj,既+adj+且+adj
Such patterns is used to connect two parallel words.


  这个苹果又大又圆。 This apple is big and round.
  敌人既凶残且愚蠢。 THe enemies are cruel and stupid.


2. Main subject (abbr: MS) + Subject's Property or Part (abbr: SP) + adj
This pattern is used to decribe the property of something.
Note: English translations in the parentheses are more formal, I think.


  他形容*憔悴。 His appearance is emaciated. (He looks emaciated)


Here, MS=他,SP=形容 and 憔悴 is the adj.
*形容 is a noun here, which means "the appearance of, the look of [a man]". Dont' be confused with "形容 (to describe)" as in "形容词 (adjective)".


  他工作能力强。 His work ability is strong. (He is capable of doing the work)


Here, MS=他,SP=工作能力 and 强 is the adj.


  我浑身无力。 My all body is feeble (I am feeble all over, from head to foot)


Here, MS=我, SP=浑身 and 强 is the adj.


3.Subject+Adjective+"了"
This pattern is used to express that the subject has been "adj" or has become (turned) "adj".


  树叶红了 The leaves has turned red.
  她漂亮了 She has become beautiful (Implies: she was not beautiful in the past but she is beautiful now)
  


4. Why do we need "很" here?&lt;
fefe thinks that this is related to the length of syllables, but I don't think so. Because even with a two-character adjective, we still could not use the consctruction "Subject+Adj" to form a sentence (except for the cases I listed above). You won't say "她漂亮", though "漂亮" has two characters. Instead, you would say 


  她很漂亮 or 她非常漂亮 (note 非常 has two characters)


In classic Chinese, ”是" was seldom used as "to be" and an adjective could be used as a verb or predicate directly. I really don't know why "很" or such adverbs are needed, but that just happens in Chinese.


  春风又绿江南岸  from the poem of 《泊船瓜洲》 by 王安石
  绿: [adj]green. Here: to make something green
  
  亲贤臣，远小人，此先汉之所以兴隆也。 from 《出师表》 by 诸葛亮 
  远:[adj] remote, far. Here: to keep oneself from, to be far away from


I just read 《史记·淮阴侯列传》 (the biography of 韩信) and I found some good examples with adjectives in classic Chinese.


  信亦知其意，怒，竟绝去。 In modern Chinese, I would say,
  
  韩信也知道她的用意，很生气，最终离开。 Hanxin also understood her intention, so he became angry [at her] and left at last.
  
  信数(shù)与萧何语，何奇之。
  
  韩信与萧何交谈几次后，萧何[对他的军事才能]感到惊奇 After several talks with Hanxin, Xiaohe got suprised [with Hanxin's military talent]
  
  见信死，且喜且怜之。
  
  [刘邦]看见韩信已死，又高兴又同情。 [Liubang] found Hanxin was dead; [he was] happy and sympathized