Is Anapanasati a complete path? Does it, by itself, lead to nirvana? I have been doing meditation of various forms (Goenka body-scanning, anapanasati, just sitting, kaya, citta, and vedana nupassana, etc.) but am trying to figure out something about anapanasati in particular.

My question is partly about Buddhist history, partly practical.

If I understand the story of Buddha's enlightenment process, he began by practicing anapanasati and attained to the various mystic absorptive states, including the eight jhanas. Yet, he failed to attain insight into dukkha, annica, and annata in any meaningful way, and remained in samsara. Eventually, after relaxing self-mortifying ascetic practices, he resolved to become enlightened, and accordingly practice anapanasati--again. 

What changed in the end? Wasn't Siddhartha practicing anapanasati all along? Why did it suddenly lead to nirvana after years of mere dead-end concentration.

Practically speaking, does anapanasati in its purest form--that is, simply focusing on the breath to the exclusion of any and all other thoughts, sensations, etc.--actually lead to the deep and abiding understanding of annata, annica, and dukkha?
Initially the Bodhisattva was practicing a form of Pranayama in the form of holding or controlling one's breath. [Deva Sangarava Sutta, Maha Saccaka Sutta]

The Breath meditation practiced later was was observation of phenomena. Each 4 Sathipattana is covered in sets of 4 Tetrad with a total of 16 steps. Also it is the complete path. The 16 steps can be found in Anapanasati Sutta and Anapanasati