It is not actually true, according to 日本年金機構 (Japan Pension Service) it states:

厚生年金保険が強制的に適用される事業所は次の事業所です。

1. 常時従業員を使用する株式会社や、特例有限会社などの法人の事業所
2. 常時5人以上の従業員を使用する個人事業所（旅館、飲食店、理容店などのサービス業は除きます 。）
3. 船員が乗り組む一定の条件を備えた汽船や漁船などの船舶

この場合の従業員とは、臨時使用人や季節的業務に使用される人を除き、厚生年金保険の被保険者とならない7 0歳以上の人を含め、労働時間及び労働日数が就業規則などに定める一般社員の概ね4分の3以上ある人をさし ます。

また、強制的に適用される事業所以外でも、事業主が、被保険者となる従業員（※）の半数以上の同意を得て厚 生労働大臣の認可を受ければ、厚生年金保険が適用される事業所になります。

（※）被保険者となる従業員とは、正社員、契約社員、パートタイマー、アルバイトなどの名称を問わず、労働 時間及び労働日数が就業規則などに定める一般社員の概ね4分の3以上ある70歳未満の人をいい ます。

Flight attendants come under "3." on the category of crew member, and I am sure there is not flight attendants at ANA with age greater than 70.

Which indicates the regardless of the title of “permanent employee,” “contract employee,” “part time employee,” etc. all employees who have work hours and work days of more than ¾ of “regular employee” are subject to 厚生年金 (government pension plan) mandated through employers. Individual employers have to define “regular employee” status on its own company accordance to Japanese labor law and have information accessible to all employees.

As indicated by ANA where contract flight attendants have less flying hours (work hours) than permanent hired flight attendants. Also contract flight attendants are not hired through an agency, contract flight attendants are hired directly by ANA.

So what boils down to is that if contract flight attendants of ANA has less than ¾ flying hours (work hours) than permanent hired flight attendants then ANA is not mandated to participate in contract flight attendants’ 厚生年金 (government pension plan). In such case contract flight attendants can be in a position that they are fully responsible (pay 100% premium) of the government pension plan out of their own pocket.

Also, it indicates that if more than half of "regular employee" has agreed and obtained a permission from the Minister of Health, Labour, and Welfare then exception to above can be made.

The link you have posted describe the definition of “contract employee” under Japanese Labor Law. However, it does not say anything about the eligibility of 厚生年金 (government pension plan). The eligibility of 厚生年金 (government pension plan) is described by 日本年金機構 (Japan Pension Service).