From the 2020 coronavirus pandemic in Japan Wiki:
"The main concern of the Japanese medical establishment was an overcrowding of hospitals by uninfected patients with light cold symptoms who believed that they had COVID-19. Medical representatives claimed that such a panic would strain medical resources and risk exposing those uninfected patients to the disease itself."
"On 17 February, the Ministry of Health released its consultation guidelines (Japanese: 新型コロナウイルス感染症についての相談・受診の目安について) to each of the municipal and prefectural governments and their public health centers. The document instructs doctors and public health nurses who staff the consultation centres to limit consultations to people with the following conditions: (1) cold symptoms and a fever of at least 37.5 Celsius for over four days while taking antipyretic medication; and (2) extreme fatigue and breathing difficulties. The elderly, people with pre-existing conditions, and pregnant women with cold symptoms can receive consultation if they have had them for two days. The guidelines also note that people who are dissatisfied with their consultation results can visit one of the specialized outpatient wards (帰国者・接触者外来) for further talks."
"After reviewing and discussing the existing data on the disease, the committee stated that universal PCR testing was impossible due to a shortage of testing facilities and providers, and recommended that the government instead limit the application of available test kits to patients that are at a high risk of complications in order to stockpile for a large cluster. Participants also noted that Japan's medical facilities are vulnerable to "chaos," elaborating that several of the hospital beds and resources in the Tokyo area are already preoccupied with caring for the surge of 700 infected patients from the Diamond Princess. They reiterated their warning that a rush of alarmed uninfected outpatients with light symptoms of the disease could overwhelm hospitals and turn waiting rooms into "breeding grounds" of COVID-19."
The above clearly indicates a strategy to limit testing for two basic purposes: 1) deal with the shortage of testing facilities and providers (as you pointed out); but also 2) to avoid a "rush of alarmed uninfected outpatients with light symptoms of the disease could overwhelm hospitals and turn waiting rooms into "breeding grounds" of COVID-19."
It isn't necessarily an either or situation here.