No, that's not the real reason.

In the places where they have shut down schools, they can't really do it either. But when government says "do X" and has good reasons, the institutions comply. I'm in the board of a private institution in a country with a shutdown order, and there was really no discussion between the board members that this institution would shut down too, despite there being no legal backing at all.

That seems to be exactly the reason in Germany. The federal government has no say regarding public gatherings or schools. They have given out recommendations, some have followed, others have not. There's no repercussion for cities not following the government's decision.

You can hide behind "I lack the legal authority" or you can act. This is one of the rare occasions that absolutely do call for the old "ask for forgiveness" mentality. Nobody in the lower tiers would complain about central government to take a very difficult decision off their plate. (some people will complain, those who still think they can just pretend that the epidemic does not happen, but they will complain anyways)

The difference is that, in China, if you don't comply the state security detains you and you get sentenced to re-education camp at best.

In other places there are no real consequences for you, even if people die. But there are legal and economic consequences if you shut down without top cover. That is why the government and legislative paralysis is so damaging -- with no safety net, people will prefer the unknown loss over the certain loss.

They do end up complying. But it's not as fast and efficient as in China.

Not saying it's good or bad, but it's definitely a different system.

This is correct afaik, I'm pretty sure the decision to close schools is with the ministries of education of each German state, however if federal institutions put out strong recommendations, state and local administrators will likely follow very soon after. Federal ministries may not have absolute control here (unless they declare a state of emergency or something) but they do have plenty of influence.

Japan has the same limitation, but the government put out a "strong recommendation" for schools to close and all the local governments complied.

Japan has a different culture. When the government puts out a "strong recommendation", the society generally follows. This even happens with corporations. There's a culture of conformity there that just isn't present in western nations.

They haven't got laws on the books to do this? - the UK Passed some enabling legislation in Jan I believe

We do, ever since the cold war.