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I guess (perhaps I'm wrong) that humans and life in general tend toward homeostatis -- i.e. wanting to stay the same as before -- and not just physically but socially and mentally too, e.g. to keep what you have and avoid what's new.

An "avoiding extremes" doctrine might encourage that.

The theory of 'flow' in psychology suggests that people want to engage in activities which are not too boring but also not too challenging.

I think that school-teachers are aware of that phenomenon, and school-children become trained (habituated) to it.

But however useful or pleasant that may be, perhaps that (i.e. the pursuit or maintenance of that 'flow' state) is just samsara.

Does Buddhist doctrine challenge this? Or endorse it, use it?

Wikipedia says,

In positive psychology, a flow state, also known colloquially as being in the zone, is the mental state of operation in which a person performing an activity is fully immersed in a feeling of energized focus, full involvement, and enjoyment in the process of the activity.[1] In essence, flow is characterized by complete absorption in what one does, and a resulting loss in one's sense of space and time.

Named by Mihály Csíkszentmihályi in 1975, the concept has been widely referred to across a variety of fields (and is particularly well recognized in occupational therapy), though the concept has existed for thousands of years under other names, notably in some Eastern religions, for example Buddhism.

The flow state shares many characteristics with hyperfocus. However, hyperfocus is not always described in a positive light. Some examples include spending "too much" time playing video games or getting side-tracked and pleasurably absorbed by one aspect of an assignment or task to the detriment of the overall assignment. In some cases, hyperfocus can "capture" a person, perhaps causing them to appear unfocused or to start several projects, but complete few.

Can you comment on either of the highlighted statements from a Buddhist perspective?

Also perhaps this is related to my previous question, Aversion and Mahayana -- I guess that 'flow' might help you to do things, even useful things, even things you might not do otherwise, like your maths homework at school -- I'm not sure it's a useful tool (or habit) for handling aversion however, i.e. any activity which takes you "out of the zone" is something you might avoid.

This sounds like the application of the practice of mindfulness in daily life.

For example, from "Chapter Six: Daily Life" of Yuttadhammo Bhikkhu's book "How To Meditate":

When cooking, cleaning, exercising, showering, changing clothes, even
on the toilet, one can be mindful of the movements of the body
involved, creating clear awareness of reality at all times. This is
the first method by which one can and should incorporate the
meditation practice directly into ordinary life.

From the teachings of Zen master Thich Nhat Hanh, we find the use of mindfulness in daily life for example:

In the kitchen:

The kitchen is also a meditative practice space. Let us be mindful
when we are cooking or cleaning in this space. Let us do our task in a
relaxed and serene way, following our breathing and keeping our
concentration on the work. A few words only may be needed about the
work at hand. We might like to start our work by offering incense to
the kitchen altar, to express our gratitude and to remind ourselves to
work mindfully.

Let us support the kitchen teams by not disturbing this meditative
space unnecessarily. We do what we have to in silence and leave the
kitchen so the teams can do their work.

While cooking, we allow enough time so we will not feel rushed. Let us
be aware that our brothers and sisters depend on this food for their
practice. This awareness will guide us to cook healthy food infused
with our love and mindfulness.

When we are cleaning the kitchen or washing our dishes, we do it as if
we are cleaning the altar or washing the baby Buddha. Washing in this
way, we feel joy and peace radiate within and around us.

Drinking tea (tea meditation):

Tea meditation is a time to be with the Sangha in a joyful and serene
atmosphere. Just to enjoy our tea together is enough. It is like a
“good news” occasion, when we share our joy and happiness in being
together.

At times, when we are drinking tea with a friend, we are not aware of
the tea or even of our friend sitting there. Practicing tea meditation
is to be truly present with our tea and our friends. We recognize that
we can dwell happily in the present moment despite all of our sorrows
and worries. We sit there relaxed without having to say anything. If
we like, we may also share a song, a story or a dance.

Eating together:

We should take our time as we eat, chewing each mouthful at least 30
times, until the food becomes liquefied. This aids the digestive
process. Let us enjoy every morsel of our food and the presence of the
dharma brothers and sisters around us. Let us establish ourselves in
the present moment, eating in such a way that solidity, joy and peace
be possible during the time of eating.

Eating in silence, the food becomes real with our mindfulness and we
are fully aware of its nourishment. In order to deepen our practice of
mindful eating and support the peaceful atmosphere, we remain seated
during this silent period. After twenty minutes of silent eating, two
sounds of the bell will be invited. We may then start a mindful
conversation with our friend or begin to get up from the table.

Upon finishing our meal, we take a few moments to notice that we have
finished, our bowl is now empty and our hunger is satisfied. Gratitude
fills us as we realize how fortunate we are to have had this
nourishing food to eat, supporting us on the path of love and
understanding.

Service meditation:

To participate in service meditation can be a great happiness. It is
an opportunity to engage in the maintenance and care of our practice
center while enjoying our practice of mindfulness. When we wash the
cars, or turn the compost piles or chop wood we stay mindful of our
breathing and the activity that we are doing. We speak only when
necessary and about the work at hand. We can maintain a light and easy
feeling as we work. An environment that is quiet can make the work
more pleasant and enjoyable. When we work in the garden we get in
touch with the plants and nourish our connection to the earth we are
living on. Sweeping and mopping the meditation halls we see that we
are already practicing to calm our mind and body. Please, do not be in
too great of a hurry to get the job done. Our most important
contribution to the Sangha is to maintain our practice of mindfulness.

Service Meditation links us to our everyday life, both here and when
we return home. As we are working at our computer or preparing dinner
for our family or teaching a class, we can practice stopping, calming
and refreshing ourselves with our conscious breathing. We can relax
and smile at our co-workers and pace ourselves to maintain a light and
serene state of being.

Most of the human actions/habits are to suppress their suffering or cover it up. The ordinary human mind's direction is avoiding certain thoughts, feelings and emotions and to do that developing certain habits, addictions. Playing games, studying the lessons, watching television, listening to music, sensuality, following the politics, watching football, talking with others etc.. Some of the human actions leads a person to a kind of "trance state" but I think that is generally not high awareness but it leads people to kind of a unconscious state that can bring him/her temporary calm. I am not sure the "hyperfocus" that is mentioned in the article is always meditative hyperfocus. In some activites like playing games, using the technology, reading a book, studying the lessons etc. it can be the opposite(because the person is not having clear awareness) unless there is some real mindfulness involves in the "focusing".

In Buddhism, meditation and mindfulness this is useful and beneficial. Absorbing in what you do in the present moment mindfully is creating a "flow" that carry you to peace, happiness and freedom from suffering. The other answers are very good so there is not so much for me to add to this subject.

Great question. Thank you.

This is such a wonderful topic for a lengthy discussion that SE does not support with its restrictive Q&A format. So now one has to perform the herculean attempt to frame an answer that might capture at least some of that broad discussion.

Your note on homestasis resonates deeply with the following from MN121

There is only this that is not emptiness, namely that associated with the six sense fields dependent on this body and conditioned by life.

In fact all of MN121 explores, if you would, the homestatic evolution of emptiness that culminates with this invitation to Ananda:

So, Ānanda, you should train like this: ‘We will enter and remain in the pure, ultimate, supreme emptiness.’

Note that this invitation was extended late in life.

"Ānanda, these days I usually practice the meditation on emptiness."

I trust I properly heard, learned, attended, and remembered that from the Buddha?

Before that came the entire Noble Eight-fold Path with all of the practice detailed in other suttas. But that's a topic for a different day.