The following two quotes are analogs in the parallel suttas MA81 and MN119 AKA Mindfulness of the Body. There is an interesting difference between them. That is, in MA81, the result of mindfulness of the body is not simple deep concentration. In MA81, the result is to know things as they really are. This is not obvious in MN119.
This seems strange at first. Mindfulness of the physical body, does not seem to relate to much. At a first blush, one thinks of it being mindfulness of tactile sensations and proprioception. This, however, ignores that kāya can also refer to the experiencer of the six senses. See bolded texts in the quote below from the Online Pali English Dictionary.
Once we see this connection to the six senses, we can then see the connection of calming or stilling “physical” actions or processes to stilling or calming diṭṭha, suta, muta, and viññāta. Where they refer to the action (perception) of the 6 senses. See bolded text below.
When we understand the “calming of physical action” (MA81) and “stilling physical processes” (MN119) to be calming or stilling of the perception of the six senses we can see that that brings about the removal of the embellishments of “the whole range of the cognitional & apperceptional faculties” (See below) and it becomes clear how this would result in seeing things as they truly are.
This now connects Mindfulness of the Body practice with the training training mentioned below to see things as they truly are and to end suffering. Notice the bolded words below.
It is more than just unfortunate that the ambiguity created by the word kāya obscures what the actual meditation practice of Kāyagatāsati is and its direct connection to liberation as given by the Buddha to Bahiya. I think it is tragic. Out of all the things the Buddha could have said to put his dhamma in a nutshell, this practice and its connection to this result was what he chose.