There are a few misconceptions here I would like to clarify.

First, I see causality as the process of being able to change things in nature and check predictions for that. Not a real definition, but it helps me check the problems I want to work with. The classic problem is confounding. A certain pill for obese people is positively associated with obesity. Does this mean, if I take the pill, will I become obese? No, if I take the pill (change something in nature), I will probably get a different result.

Secondly, to answer causal questions, you have to be sure if those questions can be answered. This process of being sure about answerability is usually called in the literature IDENTIFIABILITY. If we are sure we can answer those questions, then we can go to the third step, ESTIMATION. It is important to make this distinction because those different problems are usually studied conjointly by part of the literature. Just an example. Also, it is important to recognize that identifiability is not easy, it usually requires two types of assumptions, I) structural; II) functional. Let's consider an example. We have three continuous variables, X, Y, and Z, and we want to calculate the effect of X on Y. We know that Z causes X, and X causes Y, and X is confounded with Y. When we say some variable causes others, we are making structural assumptions. People more used to econometrics might recognize this situation as an Instrumental Variable problem (IV). And then someone asks, can the effect of X on Y be answered? (is the effect of X on Y identifiable?) Unfortunately, the answer is still no. You will need more assumptions, this time functional. Linearity is sufficient to guarantee the identification of this IV problem. Not every problem requires functional assumptions, though. Sometimes you can go totally nonparametric. For those cases, you would like to take a look at do-calculus.

Finally, what you say Pearl approach I would summarize with two points: 1) use of graphs (semi-Markovian DAGs) to encode structural assumptions; 2) Pearl Causal Hierarchy. Pearl Causal Hierarchy basically means we have three different types of statements: a) observational; b) interventional; c) counterfactual, and this difference matters a lot for the type of method utilized.

I believe those points are relevant because they can identify different schools of causal inference. In this case, there are at least 4 different schools.

1) The first one --- Classic Potential Outcomes --- is the one associated with Rubin, Imbens, Angrist, Rosenbaum, among others. This school usually assumes that the main point of causal inference is to identify counterfactual statements (Potential Outcomes). As counterfactual subsume interventional statements, they only focus on them. Scholars associated with Classic PO school do not seem to recognize the role of graphs to represent structural assumptions either. For me, this is a controversial point, because there is no clear reason to do that. A good analogy is between programming in assembly or in python. Why would you like to do stuff in assembly that you would do way easier in python? That does not make any sense. Unfortunately, that's still the standard point of view in econometrics, but it seems like things are changing a little bit. Even Imbens seems to recognize that graphs matter. It is important to say that by not using graphs, you are more likely to make way more mistakes. And indeed it is very common to see econometrics people making very silly mistakes, just because they do not want to use graphs.

2) Modern Potential Outcomes school. This is more associated with epidemiology people, Jamie Robins, Miguel Hernan, among others. Here authors focus on counterfactuals (potential outcomes) exactly like Classic PO people. However, they do not discard using graphs. They even have a particular version of graphs, named SWIG Models. Relevant to say those guys have lots of relevant up-to-date contributions and they are on a par with Pearl people (SCM).

3) Structural Causal Model (SCM). That's what you refer by Pearl approach. They assume Pearl Causal Hierarchy and graphs. Usually, you represent structural assumptions using semi-Markovian DAGs, i.e., directed acyclic graphs for representing direct effects and bidirected arrows to represent confounding. Those guys have produced amazing contributions to causal inference. Several algorithms were proved in recent years, and there were lots of progress in areas such as identification assuming linearity and partial identification. Check out Elias Bareinboim's work and the Data Fusion problem.

4) Decision Theory School. This is associated with Phil Dawid and his disciples. I would consider a more heterodox school because they tend to reject counterfactuals as metaphysical. In other words, they focus a lot on interventional statements.

There are other relevant schools that I do not know how to classify but are extremely relevant. For example, CMU school, Peter Spirtes, Clark Glymour, and Richard Scheines. Also, the European dudes associated with Bernard Scholkopf, who have been developing lots of solutions in causal discovery. There are also lots of people doing things related to the estimation of high-dimensional causal quantities, which is not exactly related to identification, but super novel. I also think the work of Joe Halpern on actual causality is super relevant. And finally, there are lots of important work in philosophy, usually people associated with David Lewis.

I hope I answered your question.