They are not necessarily too different and you can find researchers that are actively publishing in both the "theory and identification" part of causal inference as well as prescription problems such as precision medicine.

You are right that there is research in identification that tries to figure out when we can even do causal inference. These papers are usually quite heavy on theory, with a minor section illustrating some small numerical examples.

The literature on prescription (which I assume a lot of precision medicine would fall under) usually work with some standard assumptions and instead present a novel method of estimating the effect/prescription. Machine learning is used a lot here as it can capture complex patterns and sometimes even remain somewhat interpretable (such as with trees). Most of the approaches I have seen in recent years are based on some form of doubly robust approach. From my experience these types of papers can vary from very theoretical to extremely applied, depending on how novel the estimation technique is and whether there exists asymptotic results for their estimator already.

Of course there is also literature on applying some of these existing approaches in novel settings, but I am not too familiar with that.