I don't think personal testimony has no worth.
But it is liable to logical fallacy and cognitive bias.
But I'll come back to that, if I may.
Wikipedia has 3 mentions of observation in it's epidemiology article.
The first is the title of a book published in 1662 by John Graunt.
With the second and third uses observation is part of the word observational and relates to the type of study design.
This study design is not defined by the collection of personal testimony.
Off the top of my head, I can't think of one which is designed like that.
Observational studies are studies which do not involve the investigator controlling the subject assignment, as in who is in the control group and who is the intervention group.
The Wikipedia article also has a section on causal inference.
This is the part of epidemiology which deals with moving from detecting correlation to demonstrating causation.
I think you accidentally truncated a sentence from there when paraphrasing and in doing so corrupted it's meaning.
You wrote:
"It also says that epidemiology can only show that an agent could have caused a problem, not that it did."
The closest to this is this sentence from the legal section (not really the best definition to use when talking about scientific terms but never mind):
"Epidemiological studies can only go to prove that an agent could have caused, but not that it did cause, an effect in any particular case."
That last bit (which I've put in bold) is fairly important when discussing causation.
Coupled with earlier parts of that section like Bradford-Hill criteria and discussion of "necessary, sufficient or probabilistic conditions" I think the Wikipedia strongly supports the idea that epidemiology can tell us about causation.
And examples of it doing so are myriad.
I'll mention only one to avoid this post getting even longer.
Smoking.
And it's relation to lung cancer, heart disease and macular degeneration.
I don't know whether those who doubt epidemiology believe these relationships to be real.
But it's the same methodology in use.