Coma was missed in (6). That's why TypeError was about int. Silly mistake. The solution was found during the writing process:

sum([(1,2,3),(4,5),(6,)], start=())

(1, 2, 3, 4, 5, 6)

itertools.chain is also well known solution, but without sum function

from itertools import chain

tuple(chain(*[(1,2,3),(4,5),(6,)]))
# or
tuple(chain( (1,2,3),(4,5),(6,) ))