What is the function of 得 in words like 覺得、記得、顯得、值得、懂得、免得、認得…? First of all, does 得 have the same meaning in those examples?
I'm familiar with 得 in other senses like 得到 and 聽得見, but those meanings seem distant.
Is it more related to usage like “我唱得不好”? The main verb seems to have unchanged meaning. Does it make more sense to analyze the words as a main verb and 得 used like the grammar particle here?
Is it just a “filler” syllable to keep the rhythm?
得 can be used as a complement that helps indicate completion or consequence of the action (preceding verb). Its use is traceable to classical Chinese (see 10th entry here):

十三學得琵琶成，名屬教坊第一部。《琵琶行》
By thirteen, an accomplished lute player I had become; In the imperial registry of the first section my name be found. (translation)

You master something if you learn it to completion. It is easy to appreciate this rule in most of the words you listed: e.g., when you first attempt to recognise (認) someone, and then you really did, you say 認得. In 免得, consequence is stressed. Others may be more elusive, because 得 is often too grammaticalised, and that modern Chinese has the proclivity of using bisyllabic words (雙音節詞).

得 in 得到 is itself a verb meaning 'to get'. 得 in 聽得見 is a potential complement. 得 in 跳得高 is a structural particle that is used to precede the adverb. They are different from 得 in point 1.

Why is it important to make the difference? That is because certain V+得s can have different meanings, depending on context. Take 記得 as an example:




得 as
Sentence
Translation




Structural particle
就這件事，我記得很清楚。
Regarding this matter, I remember very clearly.


Complement for completion/consequence
我記得這件事。
I recall this matter.




Notice how the first 記得 precedes an adverb, but how the second precedes a noun.
In modern Chinese, 我記這件事。 is unidiomatic (bisyllabic proclivity). When you begin to recall something and you did recall it, you say 記得 (indication of completion). Both justify the use of 記得 over 記.