I think the easier answer is that asthma was not really universally seen as a psychosomatic illness. To understand why, one must problematise the definition of disease and put dominant versions of what counts as disease in context.
As a side note, this is hardly exclusive to asthma - the debate over the definition chronic fatigue syndrome / myalgic encephalopathy, a debate over psychological vs physiological establishment of the chronic disease is ongoing .
Going back the case of asthma, why recruit psychosomatic definitions, then? That is most likely to be connected to the psychosomatic movement of the early ~1930s. The first half of the 20th century was a golden age for psychoanalysis. Around 1900, Freud's proposed theories had gained popularity and a psychoanalysis society was formed (which further institutionalised the approach). Some of the prominent psychoanalysts lectured in universities around the world. At this time, psychoanalytic explanations had gravitas in a clinical context.
Much like in the other diseases in the holy 7, the diagnosis for asthma stumped contemporary physicians. So, when the biomedical explanation fell short, others were recruited to cover the gap. That is not to say, however, that the biomedical definition of the disease was thrown out of the window - again, there were competing definitions for what actually constituted asthma. By the turn of the century, William Osler offered a very biomedical definition for what constituted asthma . In addition, the biomedical definition was very present in therapeutics with, for example, beta 2-agonists and theophylline having used in asthma therapy since the 1920s .
I'd also like to point out that the psychosomatic explanations were not an artifact or a bump on the road to the right explanation. Psychosomatic dimenstions continue to be an area of interest in defining asthma . Some of the ones identified in the early 20th century have been downplayed. Others, like stress, have been incorporated as triggers / exacerbating factors of the disease . In fact, even within the biomedical model, a lot has changed as to what is perceived to constitute the disease since the 1950s. It went from being a smooth muscle disease, to an inflammatory disease, to an airway remodelling disease, to a combination of all of the above . It is even now moving towards a molecular definition .