I think you are either reading too much into the apparent similarity between “kind” and 種類 or missing the peculiarity of the English word “kind”. The usage of “kind” in “a kind of X” is kind of exceptional. In fact, from the way you phrased your previous question, you seem to perceive X as the core part of it, rather than “a kind”. This clearly contradicts how the structure “A of B” is generally interpreted. English being a head-initial language, its core part, or head, is supposed to be “a kind”. (This led to the doubt in my comment to your previous question.) If native speakers of English don’t see it that way, then “a kind of X” is an exception rather than the norm.
What if you replace “kind” with another word with a similar meaning, such as “category” and “variety”? I believe X will lose its status as the head and you will understand "a category of X" more as a category than as X.
種類 works more like those normal nouns than “kind”.
絵画は種類の絵だ is at least as wrong as “a painting is a picture of (a) category”. Remember that the core part, or head, of the Japanese phrase 種類の絵 is 絵, not 種類, because Japanese is a head-final language. 種類の is supposed to modify 絵 but that's where it fails to make sense.
絵画は絵の種類だ is grammatical as it corresponds to “a painting is a category of picture(s)”. However, it still sounds strange for a totally different reason. It sounds as if being a category is the definition of 絵画 because you are essentially saying “a painting is a category”.
Xの種類 would work better with a sentence like マスカルポーネはチーズの種類だ, which would be a valid response to the question 「マスカルポーネって何だ」. Your example, on the other hand, was not meant to answer such a question about 絵画. 絵画は絵の一種だ is what you meant. (チーズの一種 also works fine with マスカルポーネ.)
The word 種類 has its own peculiarity. It is a normal noun all right but can also be used like a counter, as in 一種類, 二種類, and so on. (This is a rare phenomenon for a Japanese noun.) 種 doesn’t work this way as 二種 doesn’t normally mean “two types” but “type 2”.
By the way, 一種 is listed in dictionaries as a word in its own right.