I believe there are two different answers for the two novels you used as an example, but both are attributable to the wisdom of the time.

In 1896, psychoanalyst Sigmund Freud and neurologist Joseph Breuer published the Studies on Hysteria. The book rapidly gained worldwide popularity. "Hysteria" referred to a psychological disorder in which an idea or fantasy that had been repressed (made unconscious) by the mind found alternative expression in the body.

The Secret Garden was published originally in serial form in 1910 and seems to be highly influenced by this theory. Colin's mother died shortly after his birth. Colin's father could not bear to see Colin, who reminded him of Colin's mother. The belief that Colin would become a hunchback and die before adulthood, his perceived infirmity, and even infirmity that has physically manifested is all a result of this delusion. In the novel, Colin is repeatedly referred to as a "hysteric", indicating the influence of Freud and Breuer.

When Mary meets Colin she has the confidence to contradict him when he complains about his non-existent hunchback and his other maladies. Unlike the servants who obey Colin without question, Mary has not been indoctrinated to believe Colin is sick or to regard him with pity and morbid curiosity. Mary's ability to contradict Colin and supplant his negative delusions with a positive outlook eradicates any illness, which was a manifestation of a mental condition (hysteria) all along.

Heidi was published before the work of Freud and Breuer, and the trope is a different one. Never is Klara regarded as a "hysteric", nor is her condition regarded as being a product of delusions. While I do not recall the cause of her sickness being discussed, her inability to get well is ultimately attributed to her physical isolation from fresh air and nature.

Here I feel the influence of the pre-germ theory, "Miasma", which attributes "bad air" as the cause of all sorts of health maladies. Miasma theory is ancient, originally attributable to the ancient Greek physician, Hippocrates (460 BC - 370 BC). Florence Nightingale's "Notes on Hospitals" was published in 1859 and was highly influential. Nightingale believed in and endorsed Miasma theory, and emphasized that patients should have access to fresh air.

Though Nightingale's common sense groundwork for nursing was highly influential and overwhelmingly positive it is dated in some respects, especially with respect to Miasma theory. Miasma theory was abandoned by physicians around 1880 and replaced by germ theory, which is the same time Johanna Spyri published Heidi in 1880. Spyri seemed to accept the common wisdom of Miasma theory and that fresh air had the power to cure all sorts of maladies (including paralysis I suppose), and thus incorporated that wisdom into her novel.

Though the influence is different between the two novels, the commonality is they are both attempts to illustrate the wisdom of the Enlightenment being employed to cure human maladies. The best way to present this in literary form is for the person so cured to be an innocent child whose condition is attributable to mistakes caused by old ways of thinking, and whose cures lie in new, modern ways of thinking. Colin was afflicted with hysteria because of the influence of his father, not any problem with his spine or lungs. Klara could not get well because her parents followed old wisdom of keeping her isolated in her sickness. The cures in both cases were brought about by protagonists who acted according to what was, at the time, more modern thinking. Mary replaces Colin's hysteria with a positive outlook. Heidi brings Klara out from isolation and into nature where she can be cured by fresh air. In this respect the theme of each novel is the same, even while they rely on different theories concerning the cause of each child's sickness.

Edit: Sloppy sentence structure.

Sickly children cured by countryside are indeed ubiquitous in literature up to 50s- and for the most part that forgotten tropes that reflect real life problems- and it was much more to it than romanticisation or belief in willpower.
It's true that spawling XIX century industrial cities didn't have the best reputation, being seen as centrum of vice and corruption, but lots of concerns raised about city raised kids had very mundane basics.

It's very possible that intended reader knew someone benefiting from being send to countryside, regardless of social class.

When you reach to the memories of the doctors, especially working with poor population you will see that development of the cities correlated with rise of several diseases in pediatrics population. Seriously, that's a sad story, since even back people know that the remiedies would be very simple:

-Sedentary lifestyle- as you an see, nearly all examples here are upper class kids tucked into beds on the border of Munchausen by proxy by worried parents and caretakers. While bed rest was thought to be cure for all diseases in early XIX century we saw a rise of the doctors seeing it as a source of the problems. Magdalena Samozwaniec, daughter of the wealthy painter, in her memories reminiscences how her older sister was basically hidden from the world due to poor health until their father returned from England where he read about new methods of raising kids- they were taken to countryside, encouraged to eat simple food, taking up sports and while her sister was never completely healed (she probably had scoliosis) they both turned into lively kids.

-Malnutrition. It's hard to phantom in modern world, but as late as in 1950s sanatoriums advertised how much children put on the weight there. In working class families calories intake barely covered needs in the good days- lack of work for any of the parents meant starvation for whole family and preserving scarce resources for those who could actually work. Milk, fresh fruits, fats were often an unobtainable delicacy. Even if family splunged on milk and butter there was no guarantee that it would come in good quality- the advice about food adulteration often took large part of household manuals.
In countryside youngsters could help themselves by foraging, catching fishes, stealing straight form orchards (that become trope of itself!) or rely on wider social networks, that often were broken when family moved to city.

One might think that it would affect only the poorest but memoirs and essays devote a lot of space to discussing diet of middle and upper class kids- often feed with small portions of the very plain and processed food (because fresh and spicy was thought to be upsetting for the stomach and bad for temper).

-Rickets and rachitis florida.
Looking from first RTG can be scary- narrow pelvis that will make childbirth impossible, bow like legs that would have been broken to function properly. Lack of vitamin D and living in cramped, dark spaced with no place to run free resulted in widespread rickets that affected mainly female population in all social strata. Girls were encouraged to stay indoors and oftentimes fed worse than their brothers who from the young age could supplement the income by working outside the home (newsies, running errands etc) and use the first charities that provided free vacations on the countryside. Again doctors with more well of clientele saw it on daily basis because girls spending time idly indoors were a sign of respectability.

-Tuberculosis and other infectious diseases

Tuberculosis was a thing since time immemorial, but reached the pandemic level in urbanisation centres of XIX century, only to be kept in check after WWII.
Most of us had contact with TB only to our organisms to fight the infection- that's how it ---works in preferable conditions. Not so much when you are young, sickly, malnourished and

constantly exposed to TB due to being in overcrowded dwellings, with lungs already weekend from constant smoke and dust. In early stages of lung TB organism still has a chance to "surround" the infection and heal, but it takes a lot of effort and removal from polluted air.
And that's pulmonary TB, one of the least serious forms. Once it developed into consumption the chances were slim. Blood, nervous system, bones usually meant death sentence.
--

Bu the 1880s the efforts of the doctors and social workers gave a rise to organised camps- paid or run by charities. The concerned parents of the working class Łódź for the hot summer months send their children to contryside in exchange for help with harvest. But it was still not enough comparing the the scale of the problem

Sources:

Pierwsza krajowa lecznicza kolonia w Rymanowie : powstanie, rozwój i znaczenie : sprawozdanie komitetu za rok 1886 Antoni Żuliński

Życie prywatne Polaków w XIX wieku, volumes 1-5, 7

Children's lit scholar weighing in to talk about the fictional side of the story, as it were. Children's fiction in English really began to take off in the 18th century, and it was soon deeply influenced by the ideas of philosopher Jean-Jacques Rousseau and his 1762 book Emile, or On Education, which was itself influenced by English philosopher John Locke; Rousseau explored the notion of nature as a superior teacher. Robinson Crusoe, the story of a man surviving on his own amid nature, was first published in 1719, and became immediately hugely popular, leading to a multitude of translations, spinoffs, ripoffs, and chapbook editions (think mass market paperback but even cheaper). While it's arguable whether Defoe was intending a message about the beneficial properties of nature, Crusoe was the single book recommended within Emile. I will defer to Rousseau scholars, which I am not, on the specific footprint he leaves, but the already existing popularity of Crusoe and similar writings suggest that Rousseau's views on nature landed on audience ready to hear them. Then at the end of the eighteenth and beginning of the nineteenth century the Romantic movement in art and writing further elevated Nature as a capitalized and important, even corrective force.

However, sickly children were getting literary attention well before this. James Janeway's A Token for Children (1671, full subtitle "An Exact Account of the Conversion, Holy and Exemplary Lives and Joyful Deaths of Several Young Children") was fiction-dressed-as-fact that was hugely popular with children themselves as well as with adults (only John Bunyan's Pilgrim's Progress was more prominent), featuring many children who languished and died "with a great deal of chearfulness," along the way inspiring and humbling the less-enlightened adults about them. To modern eyes it can seem like a slaughter of the innocents, but aside from teaching what was considered important lessons of faith, it made the child the powerful and heroic center of the narrative. Two centuries later and on the American side of the Atlantic Ocean, Louisa May Alcott's Little Women (1868/9) may not have had Janeway's intensity, but Beth March, one of the four protagonist sisters, could be right out of Janeway with her illness-infused rectitude and modeling of superior piety.

Obviously a sick child is a narrative convenience in any story of healing--they've got to be sick in order to get better. But the motif of healing of sick children, whether through nature or not, draws on that long-standing trope of the sick child's narrative power. Susan Coolidge's What Katy Did (1872) and its sequels follow Katy, twelve years old when she suffers an injury that leaves her bedridden for four years. Suffering turns Katy from an impulsive, headstrong child (she was engaging in a forbidden activity when she fell and injured herself) to a young woman of discipline and thoughtfulness; she's a post-Janeway heroine in that she's allowed to heal and find joy in this world, not just the next.

It's also worth noting that Burnett's well-known earlier book A Little Princess had an element of rescue fantasy to it as well; while Sara Crewe (who first appeared in the magazine story version in 1888) suffered no illness, the story's final triumph depends on her being squashed down deplorably and unfairly, the better to rise again. The originality of Burnett in The Secret Garden is that it's the children who rescue themselves and each other, rather than their being rescued by an adult deus ex machina. On the nature front, though, the beginning of the 20th century had children's literature blending a Rousseauian/Romantic taste for nature with neopagan elements--Peter Pan (first seen onstage in 1904) has Pan in his name for a reason, and Kenneth Grahame's bucolic animal fantasy The Wind in the Willows (1908) has Pan actually make a significant if unnamed appearance, albeit in a chapter that often baffles child readers. The Secret Garden especially (as compared to Heidi) involves elements of that artistic treatment of nature as an ancient magic.

It's interesting to note Dorothy Canfield Fisher's Understood Betsy (1917) as a comparison to the novels the OP mentions. Canfield Fisher was an American writer and educational reformer (she was an early advocate for the Montessori method), close friends with Willa Cather. Understood Betsy is essentially a robust, down-to-earth American version (Burnett was Anglo-American, with time in both countries) of the nature-is-healing trope: young orphan Elizabeth leaves the care of her city aunts for her reportedly rough country cousins on a Vermont farm, and after initial adjustment thrives in the country. But she's not a tragic or sickly figure in the city, just constrained; it's a book that offers the concept of blossoming separate from that of healing, and while Colin and Clara, I would argue, become narratively less interesting when they gain their health, turning Betsy makes Elizabeth more vivid.

While more can always be said, you may be interested in my write-up on open air schools starting in the 1900s and the attempts to bring more "fresh air" to children.