I'll start by addressing bleeding and leeching together, since these two are related, the leeches being used as method of bleeding (as opposed to blade). It's important to distinguish what pre-modern people thought these techniques were good for and what their actually physiological effects are. Bleeding (along with purging) was based upon the humoral system, wherein the aim was to either correct an excess of blood to otherwise obtain a balance in the humors.

Though a variety of compounds have been extracted from leeches, their direct applications are fairly minimal, basically being used as a sort of local anticoagulant for surgeries involving smaller blood vessels. Bleeding, in general, is never used in modern medical practice, never should have been used in medical practice, and is stupid. Someone is bound to mention hemochromatosis, which is genetic disorder leading to excess iron, which is can be treated by giving blood. This is rare clinical disorder though. The practice of bleeding and purging to control humors does not even have this fig leaf.

What the humoral system did have, was a long pedigree. We can see this with Galen who, already in the 2nd Century CE had a long list of names to drop when bludgeoning Erasistratus over matters over how the humors formed:

The learned Erasistratus, however, overlooks—nay, despises—what neither Hippocrates, Diocles, Praxagoras, nor Philistion despised, nor indeed any of the best philosophers, whether Plato, Aristotle, or Theophrastus; he passes by whole functions as though it were but a trifling and casual department of medicine which he was neglecting, without deigning to argue whether or not these authorities are right in saying that the bodily parts of all animals are governed by the Warm, the Cold, the Dry and the Moist, the one pair being active and the other passive, and that among these the Warm has most power in connection with all functions, but especially with the genesis of the humours. Now, one cannot be blamed for not agreeing with all these great men, nor for imagining that one knows more than they; but not to consider such distinguished teaching worthy either of contradiction or even mention shows an extraordinary arrogance.

Seriously, Erasistatus, you're going to disagree with Hippocrates?

Bleeding and purging continued to be popular even into relatively modern times, with echoes of the humoral system, but also incorporating new anatomical knowledge. Benjamin Rush, who treated such Revolutionary luminaries as George Washington and Benjamin Franklin, published his 4 volume Inquiries and Observations in the late 18th/early 19th century, and it is a paean to bleeding. While still keeping elements of the idea of a hot/cold dichotomy, he also moves away from the more hydraulic view the humors into focusing on anatomy. Writing on his experiences, he identifies fevers, chills, and spasms as a result of over-stimulation of the body, though not of the humors. He concludes that, "the common forms of fever are occasioned simply by irregular action, or convulsion in the blood vessels."

His solution? Bleeding and purging, to restore the normal function; essentially the idea of balancing the humors evolved.

Rush was controversial even in his day, as bleeding was quickly falling out of favor, but he provides loads and loads of anecdotes about the beneficial effects of bleeding which he himself witnessed. This is not a time when the concept or the tools for meta-analyses or systematic reviews to established evidence-based best practice were available. Instead, we had a respected physician providing his own case notes to support a logical framework for why his practices worked and to explain away instances when it did not (too much bleeding, not enough, too early, too late, etc.). At the same time he could point to actual physical effects, for instance saying that he "bled in all those cases in which the pulse was preternaturally slow, provided it was tense."

While increased pulse rate is typically associated with fever, the opposite is true in Yellow Fever, and Rush had a wealth of data from his time during the 1793 Epidemic. Bleeding of the amount that Rush performed can lead to hypovolemia. And what is are some key signs of hypovolemia? Increased pulse rate and reduced blood pressure, as well as coolness of the skin. In other words, Rush was in no way treating the actual cause of disease, but he was treating the symptoms by inducing a different pathological state.

But hey, measurable, observable effects and all that. We can see the same reason for the popularity of mercury, which has a long history in the pre-modern pharmacopoeia for any number of ailments, but is well-known as a cure for syphilis. We should keep in mind that there were as many "standard" treatments as there were providers, but virtually every mercury treatment (suppositories, breathing in fumes, topical applications, etc.) would tend to provoke mercury poisoning. So again, a measurable, observable effect. That the treatment may or may not work as an incredibly crude chemotherapy is almost beside the point. The 1909 invention of a (relatively) more effect treatment using arsenic was a similar matter of the cure being poisoning yourself and the pathogen, with the expectation that you would survive and it would not.

Of course, one of the key things to know about syphilis is that its symptoms go away on their own. The disease does not disappear, it just enters a latent phase, to return later for secondary and tertiary syphilis. So what can seem like a cure can just be the natural progression of the disease. One of the most popular non-mercury treatments for syphilis was, for centuries, guaiacum, a tree native to the Americas which would be used to prepare drinks and topical applications. Yet, despite being a recommended treatment for centuries, guaiacum does absolutely nothing. Other treatments, like special diets, purgatives, and (of course) mercury could be paired with it to achieve some sort of effect, but the wood alone does nothing.

Those are just a couple examples, but you can see how not having the tools to understand the underlying cause of disease, but instead using a corpus passed down and affirmed by basic principles and abstract reasoning, combined with the natural progression of disease and the body's own physiological responses to illness and "treatment," can lead to a false sense of efficacy.

Leeching and maggots certainly had enough benefits that they are making a recurrence in modern medicine

It is important to know that premodern medicine had rather poor grasp of the underlying causes of the disease and instead relied on subjective opinions based on experience of more renowned doctors. They used techniques that usually worked. I'm at the phone right now, but apart from mentioned leeches, they did use working analgesics and even some kind of antibiotics.

They were prone to perpetuating common myths due to lack of real evidence based medicine. For starters I would recommend reading about Semmelweis, who was the pioneer of modern antiseptics and his problems.

Apart from leeching and maggots, which /u/The_Dead_See mentioned, the practice of trepanation is ancient and still sometimes used in modern medicine for reducing Intracranial Pressure in patients with Severe Traumatic Brain Injury.