With the proviso that it depends how you define the scientific method...

One strength is it gives us a reasonably reliable way to investigate and share information, moving slowly forward with problems even though the people working on them might never meet, or even be alive at the same time.

A major downside is that (at least most popular versions of the scientific method) are designed to look at population level tendencies. And depending on the design and scale of these studies it can erase genuine differences. Let say we take a 50 people with skin rashes and give them some antifungal cream. For the vast majority of people this doesn't help, and so our study shows that it's an ineffective treatment for rashes. If we'd found a group of 50 people with rashes caused fungal infection, it would have been a highly effective treatment. So, if that's the extent of our knowledge of rash treatments we would dismiss claims that antifungals "really helped me" as quack anecdotes.

Obviously, this is the process of investigation and refinement that is part of the science. But in the interim period, when working with things that we know we do not fully understand, we have to be careful to not over privilege "scientific evidence". In a relatively new field, if one approach has "good evidence" and others don't, this doesn't mean they are necessarily less effective. They might just be less amenable to experimental designs that allows for their effectiveness to be shown, or they are effective for a specific subgroup that hasn't been clearly identified yet. (obvs, this is not meant to be taken to say any woowoo bullshit 'could' work, but that there's a whole messy middle between those two extremes.)