Osaka - Kyoto - Hiroshima - Miyajima - Tokyo

Hi,
You should use the pass to get the most value out of it - that is, from long distance travel, especially by bullet train. It really is up to your itinerary. A day trip to Nikko would save you a lot more than going from Kansai Int'l into Osaka, and there is a 1000 yen bus you could use to get into Osaka anyway. If you have a mind to go, then visiting Himeji castle near Kyoto by bullet train makes an excellent side trip and adds savings to your pass. Going to Nikko north of Tokyo would increase that substantially. Spending at least half a day to see Nara is well worth it as well. You could also go see Iwakuni with its iconic historical bridge and mountaintop castle, and it's just a stone's throw past Miyajima.

To leave Tokyo for Narita, there is a 1000 yen bus too.

So, it would probably not be worth it to activate the pass just to get from Kansai airport into Osaka, correct? I should take the bus instead, and activate the pass closer to September 30 when we make the journey to Hiroshima. That way we can get a few side trips in from Kyoto and also from Tokyo.

Yes, that is correct. As for Hakone, it is of limited use for going there. It is better to get a Hakone Free Pass. You also might look at a 3 day subway pass to see Tokyo.
In Tokyo a few of the best places to see are the Tsukiji Fish Market, plus the site that every last tourist goes to see, the Sensoji Temple. If you are in Tokyo on a Sunday afternoon, be sure to go visit Harajuku to see the youth with their wild fashions. Meiji Shrine and Yoyogi Park are next to it and well worth seeing also. And be sure not to miss the Shibuya Hachiko intersection especially on a weekend when it looks like half the city is crossing the street all at once.