Sakkāya-Ditthi and Self-View
In my understanding, there is no controversy here at all. The explanation can be very simple:

Before stream-entry, someone can think: "I am this body" or "I am inside this body" or "I am the perceiver of all my experience" or "I am the decision maker of all my actions" or "I am my memories" or "I am the thinker of thoughts".

With stream-entry these mistaken views are abandoned. Body is just body. Experience is just experience. Actions are just actions. Memories are just memories. Thoughts are just thoughts. They are not "I", there is no "I" in them, nor is "I" doing them.

At the same time, after stream-entry there are still remaining hangups to be abandoned. Particularly, there are still some habits of getting drawn to pleasant experiences and avoiding unpleasant experiences, once those are abandoned there remains anxiety about not being perfect enough (udacca-kukucca), once that is abandoned there is pride of being perfect (mana). Having these issues does not require identifying anything as "I". Even without thinking "I am this" or "I am that" there are states of attraction, aversion, anxiety, or pride.

You are asking, how is it possible that someone could keep working on these issues without thinking "I am body", or "I am inside body" or "I am the thinker of thoughts" or "I am the doer of actions" etc.?

From my experience, working on them does not require thinking in terms of "I". Instead, what's required is to see "body as body", "feelings as feelings", "thoughts as thoughts" etc. - as explained in Satipatthana. When there is such seeing, there is observation: "here is a thought conducive to dukkha" and "this thought conducive to dukkha should be abandoned" and "this thought conducive to dukkha has been abandoned" - as you can see this can happen without any I-making.

At the same time, even after abandoning explicit I-making (thinking in terms of I) there is still deep-lying residual habit or intuition of assuming that "I" is the decision-maker of actions and the experiencer of experiences. As explained in the suttas, this will linger for a long time and requires very intensive and deep meditation practice to be abandoned.