A potential problem with some of my prior reasoning is that in the body section of the Ānāpānassati Sutta we find this:
“Bhikkhus, on whatever occasion a bhikkhu, breathing in long, understands: ‘I breathe in long,’ or breathing out long, understands: ‘I breathe out long’; breathing in short, understands: ‘I breathe in short,’ or breathing out short, understands: ‘I breathe out short’; trains thus: ‘I shall breathe in experiencing the whole body of breath’; trains thus: ‘I shall breathe out experiencing the whole body of breath’; trains thus: ‘I shall breathe in tranquillising the bodily formation’; trains thus: ‘I shall breathe out tranquillising the bodily formation’—on that occasion a bhikkhu abides contemplating the body as a body, ardent, fully aware, and mindful, having put away covetousness and grief for the world. I say that this is a certain body among the bodies, namely, in-breathing and out-breathing. That is why on that occasion a bhikkhu abides contemplating the body as a body, ardent, fully aware, and mindful, having put away covetousness and grief for the world."
This would have mindfulness of "‘I breathe in long,’ or breathing out long,..." occurring
after the hindrances have been abandoned but
prior to breathing in gladdening the mind, since the opening of the sutta has mindfulness of long and short breaths occurring before any training begins. Naturally this would mean there would be gladness of the mind before the active training of "breathing in and out gladdening the mind". However, a closer examination reveals that this is not necessarily the case:
“Bhikkhus, on whatever occasion a bhikkhu, breathing in long,
understands: ‘I breathe in long,’ or breathing out long,
understands: ‘I breathe out long’; ..."
"Yasmiṃ samaye, bhikkhave, bhikkhu dīghaṃ vā assasanto ‘dīghaṃ assasāmī’ti
pajānāti, dīghaṃ vā passasanto ‘dīghaṃ passasāmī’ti
pajānāti;"
Pajānāti being comprised of the prefix "pa" which adds emphasis to "jānāti", or knowing/understanding. In other words, he really knows this. He has clear understanding and insight of this. Based on this reading then it would be possible to have the following situation:
Sits down > establishes mindfulness > mindful of long/short breath > breathes in abandoning the hindrances and gladdening the mind > ... pajānāti after all of ānāpānasati has been developed in full.