That's one way of doing it in Jap. When I mentioned upthread about their "interleaving of two different number systems", that's what I was sperging about.
One of their counting systems is derived from Chinese, and that is the very logical, simple one you've referenced in your post, but they also have a native, Japanese one as well. Sometimes, there is a choice of which to use, other times, one system only is used, while still other times, either system can be used, but with different meanings.
E.g.:
一日 ("One" "Day") can be read as either "ichi-nichi" (Chinese-style) or as "hito-hi" (Japanese-style).
二個 ("Two" "Objects") is read as "ni-ko", while 二人 ("Two" "People") can be read as either "ni-nin" (Chinese) or, more commonly, "futari" (Japanese).
Their words for 20 are extra-retarded, as using that number as a simple integer is "ni-juu", as you wrote above, BUT! twenty years of age is "hatachi", and all ages other than 20 end in the counter-word "-sai" (歳).
Calendar dates from 2~10 end in the Japanese counter word "-ka" in combination with Japanese integers (2 "fu", 3 "mi", 5 "i", 6 "mu", etc.), while dates from 11~31 use Chinese integers (1 "ichi", 2 "ni", 3 "san") with the counter-word "nichi", BUT! the fourth is "yok-ka", while the fourteetnth is "juu" (10, Chinese-style) + "yok-ka" (4, Japanese-style), and the twenty-fourth is "ni-juu" (20, Chinese-style) + "yok-ka" (4, Japanese-style). The first of the month alone has a special name: "tsuitachi" (from "moonrise" 月立).
There is similar retardation for dates ending in 8 (yoo-ka), and the twentieth alone is — for some reason — "hatsuka", because fuck you, that's why.
Oh, and the 30th day can also be read as "mi-so-ka", from the Japanese-style 3, and the Japanese-style 10 (but 10 itself isn't "so", but "too"; only higher multiples of 10 are "so").
100 is usually Chinese "hyaku", but sometimes Japanese "momo", while higher multiples of 100 (200, 300, etc.) are "o", as in 800 "ya-o", and while 10,000 (万) is almost always Chinese "man", it's Jap-style "yorozu" as in 万屋 "yorozu-ya": general store.
And people wonder why the Japs are so neurotic and drink so much.