Concentration doesn’t by itself lead to an inability to suffer. Check MN 70, where Gautama lists “seven (types of) persons existing in the world”. In particular, the third person has “apprehended those peaceful Deliverances”, meaning the arupa jhanas, and yet has not completely destroyed the cankers:
And which, monks, is the person who is a mental-realiser? As to this, monks, some person is abiding, having apprehended with the person those peaceful Deliverances which are incorporeal having transcended material shapes; and having seen by means of wisdom some (only) of his cankers are utterly destroyed…. This, monks, is called the person who is a mental-realiser. I, monks, say of this monk that there is something to be done through diligence….
(MN 70 “Kitagirisutta”, tr. PTS 478-480 pp 151-154
In MN 4, Gautama recounts directing his mind in the fourth rupa jhana to “the knowledge and recollection of former habitations”, to “the knowledge of the passing hence and arising of beings”, and to “the knowledge and destruction of the cankers”. The knowledge and destruction of the cankers involved both understanding as it really is the four truths, and understanding as it really is four similar truths about the cankers.
I don’t know of any modern Buddhist teachers who lay claim to “the knowledge and recollection of former habitations”, or to “the knowledge of the passing hence and arising of beings”. Are these knowledges prerequisites to being freed through “intuitive wisdom”, or as mentioned in Satipatthana, “profound knowledge”?
One thing seems clear, the Gautamid did his directing of mind in the fourth rupa jhana, and although Satipatthana (MN 10) mentions concentration only in passing, Maha Satipatthana does include details of both the four truths and the four rupa jhanas (DN 22).
Now to return to “one-pointedness”–there is no “right concentration” without one-pointedness:
And what… is the (noble) right concentration with the causal associations, with the accompaniments? It is right view, right purpose, right speech, right action, right mode of livelihood, right endeavor, right mindfulness. Whatever one-pointedness of mind is accompanied by these seven components, this… is called the (noble) right concentration with the causal associations and the accompaniments.”
(MN III 117 tr. PTS vol III p 114; “noble” substituted for Ariyan; emphasis added)
Many, like Thanissaro Bhikkyu, take “one-pointedness” to mean a singular object of attention:
… Show your lack of contempt for your meditation object by giving it your full attention and mastering concentration… Gather the mind around its one object.
(https://www.dhammatalks.org/Archive/Writings/CrossIndexed/Uncollected/MiscEssays/OnePointed160822.pdf, Thanissaro Bhikkyu; parenthetical added)
In my experience, “one-pointedness” has more to do with the self as a singular entity than with single-minded attention to a meditation object.
A teacher in modern India, Nisargadatta, described the self as “the consciousness in the body”:
You are not your body, but you are the consciousness in the body, because of which you have the awareness of “I am”. It is without words, just pure beingness.
(Gaitonde, Mohan [2017]. Self – Love: The Original Dream [Shri Nisargadatta Maharaj’s Direct Pointers to Reality])
The consciousness associated with “I am” is at one single location at any given moment.
Nisargadatta went on to say:
Meditation means you have to hold consciousness by itself. The consciousness should give attention to itself.
(ibid)
Zen teacher Koan Franz talked about the difference between attention to a meditation object, and attention to consciousness (“the base of consciousness”), as he discussed the meaning of the zazen instruction “place your mind here”:
So (in seated meditation), have your hands… palms up, thumbs touching, and there’s this common instruction: place your mind here. Different people interpret this differently. Some people will say this means to place your attention here, meaning to keep your attention on your hands. It’s a way of turning the lens to where you are in space so that you’re not looking out here and out here and out here. It’s the positive version, perhaps, of ‘navel gazing’.
The other way to understand this is to literally place your mind where your hands are–to relocate mind (let’s not say your mind) to your center of gravity, so that mind is operating from a place other than your brain. Some traditions take this very seriously, this idea of moving your consciousness around the body. I wouldn’t recommend dedicating your life to it, but as an experiment, I recommend trying it, sitting in this posture and trying to feel what it’s like to let your mind, to let the base of your consciousness, move away from your head. One thing you’ll find, or that I have found, at least, is that you can’t will it to happen, because you’re willing it from your head. To the extent that you can do it, it’s an act of letting go–and a fascinating one.
(“No Struggle [Zazen Yojinki, Part 6]”, by Koun Franz, from the “Nyoho Zen” site)
Franz suggested that the base of consciousness can move to a location in the body outside the head, through “an act of letting go”.
Gautama also spoke about letting go, in particular about “making self-surrender the object of thought”:
… making self-surrender the object of thought, (a person) lays hold of concentration, lays hold of one-pointedness.
(SN 48.10, tr. PTS vol. V p 174)
Given a presence of mind that can “hold consciousness by itself”, activity in the body begins to coordinate by virtue of the sense of place associated with consciousness. There can come a moment when activity takes place solely by virtue of the free location of consciousness. Activity solely by virtue of the free location of consciousness is the cessation of volition in activity of the body, in particular the cessation of volition in the activity of inhalation and exhalation, the hallmark of the fourth rupa jhana.
I believe Gautama took it for granted that his audience understood the role of concentration in the mindfulness that he recommended, just as he took it for granted that his audience understood the role of “one-pointedness” in concentration. It’s a mindfulness that was Gautama’s way of living (SN 54.11; tr. PTS vol. V p 289: “the Tathagatha’s way of life”), in particular the mindfulness of in-breathing and out-breathing described at SN 54.1 and again in Anapanasati (MN 118). About that mindfulness, he said:
… if cultivated and made much of, is something peaceful and choice, something perfect in itself, and a pleasant way of living too.
(SN 54.9, tr. PTS SN vol. V p 285)
My understanding is that when suffering exists, the four truths apply. I don’t see that life per se is suffering. Observing detachment from the pleasant, the painful, and the neutral was a part of Gautama’s mindfulness, of his way of living, yet he still recommended his way of living as pleasant.