I'm going to focus on a narrow topic: what if you can't do a two group experiment, either randomized or observational? What if you have only one group? Or what if you are talking about some national policy change where, because the change happened to the entire country, there's no obvious control group? I think you can attribute causation in some limited circumstances here.
In the clinical setting, health services researchers obviously prefer to conduct randomized clinical trials where possible, and the standard is to conduct a before treatment and after treatment measurement in each arm. In a very limited number of clinical settings, we might be able to make some causal inference in single-arm studies, as discussed by Scott Evans:
...single arm trials are best utilized when the natural history of the disease is well understood when placebo effects are minimal or nonexistent, and when a placebo control is not ethically desirable. Such designs may be considered when spontaneous improvement in participants is not expected, placebo effects are not large, and randomization to a placebo may not be ethical. On the other hand, such designs would not be good choices for trials investigating treatments for chronic pain because of the large placebo effect in these trials.
In my interpretation, say you have some very severe disease. Its mortality rate is well known and pretty high. Say that we know that 80% of patients die within one year of contracting disease X. Say we have a case series (i.e. a set of cases alone, without controls) where patients were given drug Y and we observed a mortality rate of 30%. In that scenario, I think many researchers would be willing to cautiously attribute causation. It might not be viable to conduct a randomized trial. If no two-arm observational studies were available, we would probably be willing to make recommendations based off just a case series.
How does this thinking extend to other scenarios, like the national intervention I mentioned? I think that economists have encountered this scenario more. I think that there are a number of studies about the outcomes associated with Medicaid (in the US, this program provides health insurance for the poor, which is an oversimplification but it will do). The thing is, Medicaid is controlled by the states (as opposed to the Federal, or national, government). Some states expanded Medicaid earlier than others. I believe economists have used this disparity to attempt to attribute causation, but I'm less familiar with that set of methods.
In health services research, hospital checklists are a nice parallel, because of the risk of spillover. You would ideally find, say, 60 hospitals, and randomize 30 of them to start using checklists. This is very hard to pull off. You might be a researcher at one hospital. The only thing you might be able to do is a before vs. after comparison. Here, you probably would want to make the pre- and post-intervention periods as long as you possibly could. I am not familiar with the issues of causation in this sort of scenario.