Pearl has a particular pet model for causality, but there are other approaches.
I have yet to see someone actually apply Pearl's model in prominent research, but we are probably getting there.

I think the question that is really on your mind is the one of (causal) identification.

For ML researchers, this may be explained as follows:
How, and to what degree, can we find a "deep" parameter that is a general or constant relationship between our inputs and outputs?
Is this possible? When? Under which assumptions?
If we do this, THEN we can run counterfactuals with our model.

But as it turns out, to convince yourself that your model really generalizes, you will have to prove to yourself that there exists a set of general relationships (parameters) in the data - which may for example be general because they are causal parameters - and that your model is sufficient to "identify" these parameters given the data source.
You will quickly find that you need to examine both the model, and the data (and assumptions that you have about it). Generally, no "model" itself is ever causal. It's always the combination of model and DGP.

One way to have a causal model is to simply assume some physical relationship exists, and estimate parameters from data. This is called structural modeling in econometrics and scientific ML in, well, physics mostly.
You bring in prior science and knowledge - like how weather fronts move, or how prices relate to costs - and use this structure to contrain your model. If you do this well, then this may be enough to identify causal parameters! Or ranges thereof.
Here is a technical treatment of such issues:

http://fmwww.bc.edu/EC-P/wp957.pdf

Another way is to find some measure of interest that you can derive without a complete model. For example, you may be able to "identify" so-called average marginal/treatment effects of some input variable without a parametric model. Then, as long as the changes in your inputs are "exogenous" in some sense, you can get a causal parameter.

In experiments, the differences in inputs are randomized. What remains as group differences between placebo and treatment are the causal effects.

In observational data, you may look for "natural experiments". Here, some exogenous variation can be found that identifies a similar treatment effect between individuals that are affected, and those that are not. Depending on your issue at hand, further techniques may be necessary to identify causal effects.

Let's make an example:

You built your start up to a huge and successful company, made up of programming teams. As CEO, you finally think about realizing your dream: Firing every non-technical team lead (say, everyone with an MBA). What effect would this have on productivity in the short term?

Well, you have lots of data about your teams, and have certainly fired a lot of people, so you build a model.
That model tells you, that firing such team leads is super good for productivity.

However, after you do the firing, you find that productivity drops. What happened?

Your model did not identify a causal effect. In this case, the firings you have observed probably occurred because the team lead was simply a bad manager. However, that data does not identify the "casual effect" of firing team leads. The "intervention" or the "treatment" you observed was not "exogenous".

Okay, let's do more science here. Let's say you try to find out the effect of technical ability of the team lead on team productivity. You run your fancy ML models, and again conclude that more technical ability of the manager makes for a better team.
But when you implement your effective training measures (or hiring measures, for that matter), the benefit is less than you expect. Again, what happened?

Well, you might have a selection effect. Technically competent team leads may select into productive teams. Or, there is endogenity: productive teams mean that the team lead learns technical stuff, instead of managing chaos. The effect is actually reversed: good teams make leaders more competent. Your estimation - your model - overestimates the causal effect. It is not sufficient to create a counterfactual, simply because you did not find the real causal chain.

Some key words to look into that are probably simpler than Pearl's full framework (and more practical, as they are applied every day):

Natural Experiments, Instrumental Variables, Difference-in-Difference, Synthetic Controls, Matching, Heckman Selection

If you are a person into technical papers, here's an approach that combines ML with causal analysis

https://economics.mit.edu/files/12538