Level of Focus in Anapanasati

Anapana Sati and Satipathana are closely related. The way I understand your focus (Sati) is to make sure that your attention does not go beyond four frames of reference. It is important to differentiate Samatha and Vipassana practice. Your thoughts (Vitakka and Vicara only. not all thoughts) stop in second Jhana. (Samatha) Anapanasati Sutta: https://www.accesstoinsight.org/tipitaka/mn/mn.118.than.html

For perspective, here is a sutta excerpt that describes 3 of the most popular meditation topics taught by the Buddha, and note anapana is especially designated for its effectiveness in ceasing vitakka (thinking):

(4 indispensable practices from AN 9.1 excerpt) a-subhā bhāvetabbā Non-beautiful [foulness perceptions] (he) should-develop, rāgassa pahānāya, (for) [lustful] passion's removal, mettā bhāvetabbā Good-will (he) should-develop, byāpādassa pahānāya, (for) ill-will's removal, ānā-pānas-sati bhāvetabbā inhale-exhale-mindfulness (he) should-develop, vitakk-upac­che­dāya, (for) [distractive] thinking's-cutting-off,

And here for more explanation on why breath meditation is so effective at that:

https://notesonthedhamma.blogspot.com/2019/04/16aps-great-for-driving-out-thoughts.html

And part of your other question:

Should all thinking be ideally diminished?

See MN 19 and MN 20 for great examples on the role of vitakka (thinking) in first jhāna. For example, first jhana includes resolves (sankappa) and thoughts (vitakka) based on metta, karuna, etc. There's nothing wrong with vitakka/thinking inherently. The problem is that people without sufficiently developed samadhi, run away with thinking and papanca/proliferation based on confused and deluded perceptions of reality (MN 18).

I will give to you a bit of advice that a bhikkuni gave about the practice of Anapanasati. She is one who has developed and practiced continuously this form of meditation, and have achieved the greatest of all rewards and is now helping others too in this Path. She was an educated person born to a Catholic family who was raised in a Christian environment who held senior positions at work and later embraced the Buddha's Teachings. Apparently, she had the ability to see past lives of others and this led to a desire to study the Buddhist Teachings making her give up her lay life and ordain.

A couple of weeks ago I had the good fortune of meeting her in person. Thus we had the good fortune of listening to her, telling all those who have a special calling, of this higher truth. The following bit of info about the practice of Anapanasati will help you. But first remember to take the meaning of “So satōva assāsati, satōva passāsati” to mean “He maintains his mind on dhamma (what is beneficial) that should be taken in (kusala or moral) and those that should be gotten rid of (akusala or immoral)”. That is to say that ALL thinking should NOT be diminished. ALWAYS take IN the GOOD and DISCARD the BAD. Now…

• Concentrate on your breath, in breath that flows. Gently flowing through your entire body, bringing life to every cell, every single vain that needs life to function.

• Concentrate on your breath, as it flows out of your body gently taking away what ever is unwanted. See how impurities leave your body with a single breath. Slowly all impurities leave, all that should leave from the body is taken away.

• The frustations that you have mentioned are just thoughts that arise. Just try to reflect on them thru Dhamma – on those thoughts of unfulfilled desires.

• In reflection… connections that has been made in mind towards desire… some thoughts gets you agitated and blinds you with anger followed by hate… and some thoughts blinds you with delusion that follows asa shadow.

• The attachments of feelings… which are connected. Attachments of different perceptions…

• There is an impermanent factor underlining all desires as it does not stay in the same way that you so wish it to be. The desire is such that you don’t recognize that fact that, slowly but very surely it travels along the lines of suffering and pain.

• In reflecting this way, you come to certain understandings…. Certain realizations, which helps in letting go of desires and attachments.

• Excepting the change… positive decisions brings new realizations. Letting go seems the only true answer there is to happiness.

• And when you do let go of desire, hate and delusion, it finally brings you tranquility and calmness… unexplained.

Yes, once you have reached sufficient level of concentration you may continue on the breath (samatha) or investigate the mind (vipassana). In my own practice I have a preference for reaching the calmest level possible, so I simply stay with breath, but more quietly, gently, less control, just allowing it to be, no more. At the same time, I don't ignore or suppress things as they arise, i just let them arise, fall, arise, fall. In one door, out the other. It's "kind of" a combination of both Calm and Insight.