Thanks everyone for a lot of good points! A couple of details in the OP have not been addressed.
I wouldn’t look at it that way. First up, anapanassati is satipatthana, there is no doubt of that. Satipatthana is broad and may be practiced many ways, of which anapanassati is the most common method.
Secondly, the identification of satipatthana with vipassana is mostly an artefact of the 20th century Burmese vipassana schools. In the suttas, satipatthana is included within the section on samadhi, and in fact is regarded as the “basis for samadhi”.
Yes, there is some connection between satipatthana and vipassana, because every part of the Dhamma is connected to every other part. But think of it this way: in the hundreds of suttas dealing with insight throughout the collections on the aggregates, sense bases, dependent origination and so on, satipatthana is almost never mentioned. Conversely, in eighty or so suttas on satipatthana, with a few exceptions, vipassana is never mentioned.
The basic meaning of satipatthana in the suttas is “meditation”. It is what one does when one goes to a secluded place, sits down and satiṁ upaṭṭhapetvā “establishes mindfulness”, i.e. “does satipatthana”, and observes the in-breaths and out-breaths. Yes, it has a broader and more subtle sense than this, but this is the core.
Vipassanā per se does not belong under “right mindfulness” of the eightfold path, but as pointed out by thomaslaw, under “right view”.
I’ve just finished a series of four talks on the topic, enjoy!