The reason I heard for omiting tachanun (I think i read it in R” Soloveitchik’s work) is that Tisha b’av isnt a time for davening. As several pesukim say Hashem didnt accept tefilos eg eicha 3:44 ???????? ??????? ???? ???????? ?????????. This is why we omit tiskabel as well.
Tisha ba’v is a time for mourning (of course we daven shemonie esrei like we are mechuyev to)