So when the Buddha goes into more real-life detail, he tends to explain the 10-link version without avijjā or sankhārā. For example, see DN 15 and MN 38. See also SN 12.65 where he describes his pre-awakening insights.
This is the mutual conditionality of consciousness and name-&-form. These co-dependent factors support one another for existence to be manifest, and from there beings crave, appropriate, and generate renewed existence accordingly with their actions for a new birth. The 10-link version is about the samsāric experience and how it generates further existence in samsāra.
With the addition of avijjā and sankhārā, we add in the factor: where did consciousness come from before that, and what factors are underlying the craving that arises? So a being is born and craves, etc., but why? These two links explain the ‘history’ behind consciousness and what is underlying it so as to automatically turn to generate more existence.
Ignorance about the four noble truths is what conditions all of the choices (sankhārā) or activities that we do by body, speech, and mind. These activities have an energetic force that steers consciousness, and it manifests as the karmic effect on consciousness via rebirth and the inclinations to craving. This explains, as I mentioned above, how/why consciousness ended up re-established in nāmarūpa. But as you pointed out, there is a degree of recursion going on.
While the sankhārā link explains the principle of kamma more than others, it also can stand in for all the activities (incl. things like craving, insisting, grasping, etc.) sparked by ignorance that propel consciousness forward. So this avijjāpaccayā sankhārā link is kind of like a summary for all of samsāra: Conditioned by ignorance, all the activities propelling consciousness around come to be. Once consciousness is established, we get a little more detail into what those activities are (craving, grasping, etc.) as well as their more specific conditions (feeling, contact, etc.).
And with ignorance at the beginning we see that all these activities, including craving, will cease when ignorance ceases. The non-sankhārā links will cease when consciousness is not ‘planted’ again, such as contact and feeling.
So in summary: the avijjā and sankhārā factors help show the brief history for consciousness being established, they demonstrate the principle of kamma explicitly, they show the underlying principle of all the active factors involved dependent arising and how they cease, and they allow the principle that ignorance underlies craving to be explicit.
I hope this is somewhat helpful. The 12 links are really just a helpful list of factors condensing all of samsāra as much as possible and applying the structure of idappaccayatā to it to allow for investigation and insight to occur, and to help mould our actions with more wisdom. For instance, putting the conditions in for wisdom to give rise to wholesome activities and to be aware of how consciousness is propelled in this impersonal tangle of conditions.