strictly speaking, under Japanese laws, they were not "war criminals"

Chibakun: "Aren't there class A war criminals enshrined there?" Not under Japanese laws. After accepting The Treaty of San Francisco in April 1952 and according to its Article 11,

The power to grant clemency, to reduce sentences and to parole with respect to such prisoners may not be exercised except on the decision of the Government or Governments which imposed the sentence in each instance, and on the recommendation of Japan.

Japanese government, now a sovereign entity, immediately began working "to grant clemency, to reduce sentences and to parole with respect such prisoners." And many did get reduced sentences and parole. Those who were killed by the Tribunal's death sentences were officially described as "公務死" (deaths while in service under captivity) in May 1952 by the Japanese Minister of Justice. The main motive for this was, not justification of militarism, but to give a legal status so that their families (wives and children) can receive compensation. If they were criminals, their families could not get legal compensation. The efforts were mainly pushed by antiwar Socialists. So there was not much controversy at that time. Therefore, strictly speaking, under Japanese laws, they were not "war criminals". Of course, years later, some people took advantage of this change of status and gave them an excuse to enshrine "war criminals" in Yasukuni since they were legally no longer criminals.