How can we use the triple round model to understand dependent origination?

The short answer is that it is easier to understand the vital conditions since each of the 22 conditions is individually and directly verifiable through personal experience. The three rounds are organizational rather than strictly experiential. They are a framework for discussion. Personally, I favor the 22 vs. the 3 because I have found that the 22 are more useful day-to-day. For example:

I say that faith has a vital condition...you should say, "suffering".

The Buddha sums up the sutta like this directly answering the last two of your questions:

In the same way, ignorance is a vital condition for choices. … Freedom is a vital condition for the knowledge of ending.

Go look up Freedom in the sutta. The sequence is verifiable and amazing. And once your read and understand it, understanding the rounds and the wheel makes sense.

As explained in oral Mahayana tradition, the twelve Nidanas schematically describe gradual process of emerging subjective experience of the world and a sense of self.

The actual process is not linear and does not have 12 well-defined steps. Instead, 12 links should be considered as key milestones or features. The process itself is made of thousands or even millions of events. Every event follows the same basic pattern: something is perceived, recognized, identified, categorized, evaluated, either liked or disliked, either craved or avoided, then there is some action, then the resulting experience is aggregated. This basic cycle occurs thousands or millions of times - each iteration slowly, slowly contributing to the developing sense of self (and the world).

There are many ways this process can be analytically explained in order to aid understanding. The challenge any description must solve is to show how each iteration follows the same basic pattern, but at the same time, to show that the whole thing grows over time, becoming more and more clear. In the beginning, there are very vague tendencies for generalization, and then slowly, slowly, very blurry outlines begin to emerge.

Another challenge is to show how and why this process is self-sustaining and self-clarifying. What forces are at play that make the whole thing evolve. What we're trying to show is how the growing subjectivity shapes the agent-centric action and how action feeds the grows of subjectivity.

This is what the triple-round model tries to illustrate. In this model we say that basically, D.O. has three main forces at play: the force of "murkiness", the force of action, and the force of growing subjectivity. These three forces support each other and push each other forward, thereby making the whole situation evolve.

Murkiness feeds and supports the growth of primitive, confused, naive subjectivity. Confused subjectivity feeds and supports defiled activity. Defiled activity feeds and supports murkiness. The three forces feed and support each other and help each other develop.

The first round is called the round of defilement (kilesa-vaṭṭa) which includes ignorance (avijjā), craving (tanha) and clinging (upādāna).

This part talks about the force of murkiness. When we look at the 12-fold description of D.O. we can see 3 features that can be primarily associated with murkiness. The initial ignorance is murkiness in its most obvious original form. Craving is murkiness in its developed phase. Updana is murkiness in its most active phase.

The second round is called the round of action (kamma-vaṭṭa) which includes mental formations (saṅkhāra) and becoming (bhava).

This part talks about the force of action. Murkiness contributes to propensity or tendency to act and react in a certain way. On the initial stages, when things are very vague and unclear, these propensities are very ambiguous generalizations, inclining to seek some experiences and to avoid others. On the mature stages of D.O. the propensity to act in pursuit of likeable experience and to avoid its opposite gets much more deliberate, so now we can talk about "bhava" or individual purposeful planned activity in pursuit of personal plans. On this stage we actively shape our environment by achieving our dreams and making them come true - in other words we actively construct ourselves and our world.

The third round is called the round of results (vipāka-vaṭṭa) which includes consciousness (viññāṇa), name-form (nāmarūpa), sixfold sense base (saḷāyatana), contact (phassa) and feeling (vedanā). Sometimes, it also includes birth (jāti), death and the rest of suffering (jarāmaraṇa).

The remaining group describes the elements that develop to comprise our subjective experience. This is the gradually emerging experience of myself looking out, this is what we see: external objects slowly take shape, our feelings, the sense of the world and of self, and the interaction between the two.

In (protomahayana) Salistamba Sutra this is explained very similarly, except there they split the Defilement (Murkiness) factor into two separate factors: Ignorance and Desire:

... Four limbs of this twelve-fold Dependent Origination [are] ignorance, desire, karma and consciousness.

Here, consciousness [is like a] seed. Action [is like a] field. Ignorance and desire are defilement. Without these conditions, the development of the consciousness-seed does not occur. And so, the seed of consciousness grows, standing in the field of action, scattered by the wind of ignorance, watered by the moisture of desire.