Argthtjtr

I’m wondering if most monks would go 99 pct of time with no mind wandering, just focus on breath.

Case 38 of the Mumonkan An Ox Passes Through a Window

Case

Goso said, "For example, it's just like an ox passing through a
laticed window. Her head, horns, and four legs have passed through.
Why is it that her tail can't pass through?"

Mumon's Commentary

If in regard to this you are able to turn yourself upside down,
attain one single eye, and utter a turning word, you will be able to
repay the four obligations above and help the living beings of the
three realms below. If you are still unable to do this, reflect again
on the tail; then you will be able to grasp it for the first time.

The Verse:

If it passes through, it will fall into a ditch;

If it turns back, it will be destroyed.

This tiny little tail-

What a strange and marvelous thing it is!

I'll let you in on a little secret - the mind never stops wandering. Oh sure, the mind of an experienced meditator looks nothing like that of a novice, but to say that it's 99% focused on the breath is a pretty romantic view. One thing that becomes apparent over the course of one's practice is that our illusions are depthless. Unless you are the Buddha himself, there will always be imperfections in your awareness. The only difference between someone who is experienced and someone who is just beginning their practice is the nature of these delusions.

To give you an example, when one first starts their meditative practice, the small mind is pretty much it's own master and it will drag itself around in limitless ways. There is internal narration, irritation at discomfort, plagues of doubt, and an assortment of other defilements affecting the mind. The mind of someone who has been sitting consistently for, say, ten years is much more spacious. One's internal narration almost completely disappears. Discomfort more or less vanishes. Doubt has mostly been cured. Those defilements that remain are much, much more subtle. Most of them are completely beyond the realm of language. Here, imperfections in awareness are dealt with in a way comparable to a master craftsman who knows how to apply his file just so to get the wood to respond in the way he wants. Just how he could never describe the feel he uses to apply his craft, the skilled meditator simply cannot vocalize the feel they have for keeping their mind applied to a counter-point sign, for instance.

More to the point, the deeper one engages with emptiness, the more likely they are to discover a veritable ecology of subliminal mental obstacles, habitual patterns, and other phenomenon arising from the personal unconscious (e.g. those issuing from the alaya vijnana or store-house consciousness). These subtle, unconscious phenomenon merely "perfume" the experience of the novice. Their presence is so pervasive - like an odor from an old, soiled couch that lingers in a room - that they are mistaken as being fundamental to the nature of experience. For the the experienced practitioner, on the other hand, these phenomenon are seen for what they are and become the fodder for much of his practice. They are also the very things that will give rise to most of the subtle distraction he experiences in meditation.

Off the cushion, the experience is mostly the same albeit it is much, much more difficult to maintain a deep level of mindfulness with so much going on in the world. The mind is far less stable off the cushion than it is on it. Because of that, any disruption makes a much larger splash. Because there is so much going on, it is also much harder to really see the much more subtle defilements (like those arising from the alaya). One might think that they are free from disturbance, but the truth of the matter is that their mind isn't quiet enough to see the torrents raging just underneath the surface of consciousness.

In short, no matter how experienced you are, there's always that tiny bit of tail.

It depends on the monk's stage of insight. A monk can't immediately have a detached awareness that make him free from the negative effects of the wandering mind unless he already attained enlightenment in his lay life. It is possible to be a very experienced meditator but not realised Nibbana yet, so the confused mode of the mind would continue for that person, monk or lay meditator doesn't matter. Also the uncontrollability of the mind continues always, but the awareness of a person can be detached from the negative effects of the wandering mind. The important thing is not the wandering mind but how the person's relationships is with the wandering mind. If a person doesn't take his/her thoughts, feelings and emotions seriously and isn't pulled by them, then the mind can continue to wander as much as it likes and it doesn't really matter anymore.

After a person realizes Nibbana and attains enlightenment(stream-entry) gradually his awareness grows. After experiencing Nibbana many times, the disidentification process from the core of the self/ego(The core of the subconscious mind) would start for a stream-enterer, which would eventually lead a person to a really grounded spiritual state.

https://en.wikipedia.org/wiki/Four_stages_of_enlightenment

Reaching the once-returner stage makes a person's awareness detached from his mind states and his thoughts, feelings, emotions doesn't lead him to a confused state anymore. But still nothing can be compared to the bliss and happiness of Nibbana

https://en.wikipedia.org/wiki/Sakadagami

A Sakadagami's mind is very pure. Thoughts connected with greed, hatred and delusion do not arise often, and when they do, do not become obsessive.

Before achieving enlightenment, the person will continue to be in ungrounded spiritual state most of the time. This is the nature of the humanity. Also a monk or a lay meditator doesn't only focuses to their breath to practise mindfulness: https://en.wikipedia.org/wiki/Mindfulness

If the monk practises Samatha meditation, then he can achieve a no-wandering state of mind and he can have a peaceful and calm mind like the enlightened people, but generally it is not easy to achieve the goals in Samatha meditation and the person needs a good teacher to guide him.

I quote from a nice article here that explains the beauty of being detached from the wandering mind(but I will not quote the last 2 sentences about control and creating our own reality because as long as we are in a human form these two things are not possible and it is not easy to change our past karma's effects immediately):

What if We Didn’t Take our Thoughts so Seriously?

We would be able to live in the present, enjoy life at a higher intensity, without having to be busy and scared of the reality from our minds.

We would feel relief, we would be free of that mental and emotional weight that our negative thoughts put on us.

We would understand that we are not what we think, we’re not that voice in our head – we are much more than that.

We would detach from thoughts and we would become the observer.
We would understand the difference between the present, reality and the mental scenarios influenced by a disharmonious mind.

https://myexoticdream.net/en/what-would-happen-if-we-did-not-take-so-seriously-thoughts/

Not sure about "experienced monk", but from my own perspective:

This question assumes the mind is single-threaded, that it thinks about one thing at a time. So, in this question's assumed framework, the mind can either "wander" from point to point or "stay" on one point, correct? - Which is not at all what happens as you gain experience in meditation.

Instead, as your meditation develops, you realize that the mind is in fact a lot more amorphous than our narrative of it, and that there are many, many different thoughts, perceptions, currents, and ebbs going on at various levels.

So your awareness expands. It opens. And as it expands, those trains of connected thoughts that your mind used to generate still occur. But now you can't really call it "wandering" because your scope is not limited to single thing at a time.

Those things that @000 mentioned: internal narration, discomfort, doubt - all kinds of inner drama - they come from this limited polarized mind that gets attached to one thing at a time and insists on keeping black-n-white about it. When the mind learns to be less polarized, less reified, less attached to black-and-white concepts - it no longer generates the types of polarized phenomena like wandering and inner drama. It becomes more like an ocean than like a river, more "analogue" than "discrete" etc.

So it's not like it stays on breath 99% of time - rather, it includes both breath and thoughts. And since you are no longer dragged around by your mind, there is no point in trying to keep it tied to any single object - be it breath or whatever.

Now, if we want to talk about progress in meditation, there is an entirely different aspect that improves as one gets better, which is scatter-mindedness vs. clarity.

When you are scatter-minded, your mind has a lot of noise coming from various activated ideas, memories, concerns, emotions, anxieties and so on. They send their signals all at the same time. As your meditation progresses, those noises calms down and you enter what can be called clarity or lucidity. In this state, things are "clear" - meaning, you have a sort of obvious spontaneous understanding of what's what - what's right, what's wrong, what you really want, what you missed, what's the most important thing you need to do - and so on. In this state of clarity, it is a lot easier to make fast spontaneous decisions - because you have a good connection with your "guts" and - well - your mind is clear.

So, to summarize my answer to your question: an experienced meditator's mind does not stay 99% on breath, instead his or her mind is in a state of open, expansive, effortless, all-inclusive, quiet clarity.