How can mindfulness be about immovable attention?
When one gets absorbed, focused, or mindful in anything, no matter what the object is, the mental factor called “ekagghatha” takes over and make the mind latched on to it. Being mindful depends on the situation. The kind of mindfulness one has when driving a car is different from the mindfulness when reading a book, and the mindfulness needed to attain a jhana is different from those two.  

The jhana that one comes to by using the breath only would not help in walking this Path. As you correctly pointed out Buddha figured this out, and managed to break through and achieve Nibbhana. Just by focusing the mind on the breath, one can achieve only a mundane type of jhana. Also trying to be mindful by doing “walking meditation” by saying “lifting the foot”, “putting it down” would only get you to this lower level mundane jhanas.

The higher levels of Jhana that furthers the Noble Eightfold Path are achieved by supermundane “ekagghatha”  that one comes to by coming to upacara samadhi.  When listening to a Dhamma discourse or reading about a Dhamma concept one can get to this type of mindfulness.

When contemplating on bad consequences of mind-made pleasures, willingly giving up by comprehending their bad outcomes one is able to come to the higher levels of Jhana.

If one’s intention is to walk this Path, the focus should be something related to Nibbana. It could be a Dhamma concept or Nibbana itself. Focusing on the peacefulness of discarding sense pleasures can be done only when one comprehends the true meaning of anicca, dukkha, anatta. By being mindful the correct way, one can come to samma sati.