The "indefinite article" is called that because it is an article* and it is used with indefinite noun phrases. It certainly isn't a necessary part of all indefinite noun phrases: the English indefinite article is specifially a singular article, unlike the definite article, so it isn't used with plural or non-count nouns.
As you've noted, not all indefinite noun phrases have "a/an", so in most contexts the presence of the indefinite article doesn't exactly mark indefiniteness. Rather, in many contexts indefiniteness in English is indicated mainly by the absence of the definite article. The presence of the indefinite article with an indefinite common noun indicates that the noun phrase is grammatically singular and count. When the indefinite article is absent, a noun phrase is usually interpreted as plural or indefininte noncount (e.g. the difference between "a stone" and "stones" or "stone"); if that interpretation is impossible, the noun phrase will just seem to be ungrammatical (as in your example "*I made mistake").
The indefinite article does serve as a marker of indefiniteness when it appears before proper nouns that usually lack a definite article even when they are definite, like people's given names. A noun phrase like "Sandy" is definite; a noun phrase like "a Sandy" is indefinite.
*The definition of "article"
The definition of the word "article" seems to be somewhat unclear. Its application to "a/an" is traditional, not necessarily based on any deep underlying connection between this word and "the" (although they do have similarities; they are both determiners).
For example, Wikipedia says that "The articles in English grammar are the and a/an, and in certain contexts some", whereas another source refers to some as an "article equivalent". Obviously, these are not particularly advanced linguistic sources; I just am trying to make the point that at least some people seem to be confused about, or disagree about, what exactly the word "article" refers to. So you shouldn't necessarily expect the terminology in this area to have much explanatory value.
A point similar to the one that you seem to be making was made in "On the Article in English", by David M. Perlmutter, in 1970. Perlmutter says that
If this is correct, the relationship between the definite article and the indefinite article in English is quite different from what has been generally supposed. Grammarians have worked on the assumption that NP's may bear either a definite or an indefinite article, and that the two constitute some sort of opposition. If the analysis given here is correct, however, the indefinite article is simply a numeral like all other numerals, and the occurence or non-occurence of the definite article is a completely independent phenomenon.
I don't know how Perlmutter's paper was received (some parts of the analysis seem a bit far-fetched to me), and I don't know what kind of progress has been made on the characterization of the indefinite article since then; I know that I've seen other analyses that treat the indefinite article as some kind of special reduced version of the numeral "one" (which is consistent with the etymology, although that doesn't prove that it's a correct synchronic analysis).