> three tsu's
Just like you cannot accept three 羽's, you don't need to accept three tsu's. In other words, as you cannot say:
山からキジ、ハト、スズメの羽が飛んできた。 [listing the elements in a set of 羽's]
so you need not conceive of actual things for つ (a tongue-in-cheek answer is つ is a non-trivial remainder of mod 10).
I think you are having issues with the fact that the autochthonous Japanese numerals are bound morphemes. E.g.,
We say きゅう ひく さん は ろく (9 - 3 = 6), using Sino-Japanese vocabulary, but we cannot say:
*よ に いつ を かけた かず は はた に ひとしい (4 * 5 = 20), using tsu-less autochthonous numerals.
The oldest example I know of calculative expression is that of multiplication recorded in 1604 by Rodrigues, a Jesuit missionary to Japan. The expression is Yotçuni caquru
(四つにかくる), and means "to quadruple". Apparently, the Japanese language has evolved since then to express calculations of pure numbers with Sino-Japanese terms, not to refine autochthonous numerals so they can express pure numbers.