Your experience is the same as mine: I could not see my turn signal flash either. I just followed the horn chirp as the signal to move to the next tire. The light might have flashed one time very fast. In the daytime, it might be not noticeable.
Any way, the horn chirp is good enough and the procedure is easy to follow. I would not be bothered about the turn signal. It is obsolete because the procedure already tells you what tire should be next. Though, next time I would do it in night time to see if it flashes or not just for curiosity