I can't comment precisely on every Native American medicinal practice with regards to symptoms of European diseases, except to point out that pre-modern medicine was very symptom focused, rather then specific disease focused. While specific ailments were recognized and named in both European and American disease schema, it's more useful to think of pre-modern medical practices as focusing on treating symptoms rather than some underlying disease.

As for the "steam lodge to cold stream to death" train, I spent some time looking for an original source for this claim. The furthest back I got was the memoirs of Ulysses S. Grant. After describing the general layout of a steam bath he talks about its medicinal use by writing:

When a patient required a bath, a fire was built near the oven and a pile of stones put upon it. The cavity at the front was then filled with water. When the stones were sufficiently heated, the patient would draw himself into the oven; a blanket would be thrown over the open end, and hot stones put into the water until the patient could stand it no longer. He was then withdrawn from his steam bath and doused into the cold stream near by. This treatment may have answered with the early ailments of the Indians. With the measles or small-pox it would kill every time.

In the very next paragraph, however, he does note:

I never, myself, saw the treatment described in the preceding paragraph, but have heard it described by persons who have witnessed it.

That is... not promising in terms of an informed account, not the least because Grant did not arrive in the Pacific Northwest until after the measles epidemic which kicked off the Cayuse War. Anything he heard about Native American medicinal practices was not only second-hand from Euro-Americans, but also had the distortion of years and war.

I did find another incidence of "steam bath-cold water" leading to excess mortality, in a 1909 publication of the California State Journal of Medicine, wherein one Fordyce Grinnel, M.D. of Pasadena states:

The sweat-house, used by the medicine men as a valuable adjunct in the treatment of many diseases, here wholly failed, for putting a patient in a hot steam bath for a time and then subjecting him to a cold plunge, was not conducive to recovery in smallpox, and many died from the treatment.

There is no citation here though. While this is not untypical of writings of time, the fact of not being able to find a more direct source is troublesome in determine whether this acutally happened or in what frequency it happened, not to mention the actual mortality. Grant says he heard it "would kill every time," whereas Grinnel says that "many died from the treatment."

The point is that heresay from outside groups is not a particularly precise source for getting the specifics of medicinal treatments. We do, however, have a description of indigenous steam bath treatment from the post-Conquest Aztec source, the *General History of the Things of New Spain*, written in the mid-16th century. It states:

As soon as the eruption appears the patient should at once be given hot baths. If no tub can be provided hot packs should be substituted. These should be given at intervals of every three to four hours and continued as long as they do not enervate the patient. The hot bath promotes diaphoresis and kidney action; helps eliminate poisonous elements in the blood and has a quieting influence on the nervous condition of the patient. A sense of relief and comfort is also furnished that is not obtainable by other means.

Oops. My apologies, that is from a 1903 edition of some rag called the Journal of American Medical Association. It then goes on to advocate scraping off the smallpox lesions with a "sharp spoon or curette" followed by a bath of "carbolic acid, bichlorid of mercury, or boric acid."

Let me try again.

In eruptive diseases, nettle-rash, small-pox, scarlet fever, varioloid, and measles, the vapour bath may be used with benefit at any stage of the disease. If the eruption be slow to appear, applying a vapour bath will bring it out to the surface; or if there be a tendency to retrocession or striking in of the eruption, the vapour bath should be applied; or at least a moist heat should applied around the patient, and powerful stimulants given -- as the third preparation of lobelia, and stimulating injections. In every stage of small-pox the vapour bath is grateful to the patient and aids the constitution in working off the disease... Scarlet fever and measles are very liable to be followed by dropsy, running of the ears, swelling of the glands, etc., arising in part from an unhealthy condition of the skin, which may be remedied by steaming the patient two or three times....

Shit, sorry. Now I'm quoting from an 1850 medical text based in the popular theories of Samuel Thomson. Let's see if I can get it right this time.

For malignant measles: a bath at 79F, for twelve minutes, with cold affusions to the head, repeated in five and nine hours... the cold bath restores the urinary secretions, the skin becomes soft, and the temperature falls, the eruption pales, but follows the regular course.

Apologies again, that's from an 1897 medical text advocating a cold bath for symptom treatment while acknowledging no effect on disease course.

Here is the actual Aztec prescription for the use of steam baths:

And the sick there restore their bodies, their nerves. Those who are as if faint are there calmed, strengthened. They are to drink one or another of the medicines, as has been mentioned... And one who has scabs, whose body is much festered, whose body is not much covered with sores, they there have [such as these] wash. When the humor has come out, then they apply the medicine. And the pustules, are there cleansed.

You can see that the Aztec remedy is not much different from those advocated by 19th C. and early 20 C. Euro-American physicians. It's important to keep in mind that, before the systematic development of microbiology and attendant search for antibacterials/antivirals, medicine really was focused on addressing symptoms rather than underlying causes. The causes of disease were unknown, though this did not prevent speculation.

Moving to the pre-modern times of European colonialism in the Americas, the situation is even more murky, as medical professionals at the time were basically working in disease models adhering to Hippocratic/Galenic ideas of humoral disease. Prior to the introduction of variolation to Europe from the Ottomans in the early 18th century, the key components of smallpox treatment were purging and bleeding, along with a wide and highly variable pharmacopoeia of herbal remedies. The use of heat for fevers and, yes, steam baths was also a key component to the European medical corpus at this time as well. The development of "fever phobia" is a relatively recent aversion to what is a part of the body's natural immune response.

So comparing the effectiveness American and European medicinal treatments at the times of contact has its difficulties, as neither were particularly effective from a modern standpoint. They were both riddled with magical thinking about disease models, heavily influenced by theosophy, reliant upon precedent, and woefully lacking in any standardized approach to best practices. Yet, those stumbling approaches to disease could yield empirically effective practices and treatments. Assessment of efficacy is best served by looking act actual accounts how those treatments were deployed and their effect (in conjunction with modern physiological understandings). Heresay about deaths from something with no clear reason to cause death? Not so much.

As long as we are looking at vague historical attestations to medical effectiveness though, this is the testimony of Motolinia, one of the original friars to arrive in Mexico after the Conquest and who wrote extensively about Mexican practices on a number of subjects, medicine among them.

They have their own native skilled doctors who know how to use many herbs and medicines which suffices for them. Some of them have so much experience that they were were able to heal Spaniards, who had long suffered from chronic and serious diseases.