I just have a few thoughts. I know that people have very different views so I'm sure that there will be some who disagree with me.
"...don’t think that’s worth putting up with the humidity"
When I said above that you need to be prepared for this, I was simply saying that you need to be aware of this and prepare for it. Not necessarily that you need to avoid it.
There are a few things that you can do; for example, embracing the Spanish concept of 'siesta' is a great idea. In the middle of the afternoon, when it's the hottest, return to your hotel and rest for a while and make use of the aircon. Then go out again in the early evening.
Alternatively look for an activity or place to visit during the afternoon that is indoors with aircon.
As well as this, keep the places that you are going to on any one day close to each other. If there is a place that you want to see that is some distance from others that you want to see, then do that another day.
Generally get out of the hotel early and get to places as soon as they open. This way, it is going to be cooler and less crowded.
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Another thing to consider is what parts of Japan you visit. As well as the typical Tokyo/Hakone/Kyoto/Hiroshima/Osaka route, perhaps think of areas of Japan that are going to be cooler and less humid.
For example, Nagano prefecture. It's about 90 minutes on the Shinkansen to the north-west of Tokyo and it's a lot higher up in the mountains so it's a lot cooler (more like a warm English summer day).
There's lots to do and see there, temples, shrines, castles etc and more nature than you can shake a stick at.
You may have heard of the "snow monkeys" in that area that are a popular attraction during the winter. Well, they're there all year round and you can go and see them in the summer as well (although they likely won't be hanging out in the onsen like they do in the winter).
There are a lot of lovely little towns to visit with good restaurants and onsens to visit. The Telegraph did a travel article on one of them last year, Karuizawa, and here is an archive of the article
https://archive.ph/iPVD1
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"I would prefer to go in summer holidays to be able to have longer there"
I mentioned August as that was when the very cheapest flights were during the school holidays. But flights in July are still going to be cheaper than Easter.
Might I suggest flying out the day after term ends in July (or the weekend after). If you're looking at three weeks then that will give you some time there in July which is very slightly cooler than August. Then split your time between the hotter places and the cooler places like Nagano.
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"I spent a night in a hotel in Guangzhou before when flying long haul with China Eastern and quite enjoyed both the break in travelling and a tiny taster of China."
You can stay for up to ten days on a stopover in China if you are flying via places like Beijing or Shanghai and you don't need a visa. But you do have to stay in that particular area, you can't go off all around the country.
It might be worthwhile having two weeks or so in Japan from the end of July and then, on your way back, have a stopover in China for three or four days. So that way your family get to see both Japan and China in one trip.