Yes, you can make a 21 day pass pay off. Once you make a 14 day pass pay off, it is not difficult to make the 21 day pass pay off with some additional travel.
You can see this yourself with the rail pass calculator.
https://www.japan-guide.com/railpass/There are some people who say you'll make a pass worthwhile by the number of trips you take, others by the numbers of prefectures you cross, and still others by the amount of time you spend on the trains...the truth is you should not pay attention to any of that - you should add together the actual fares and see if it is worth it - that's the only way to know for sure, without any ambiguity.
There are some points on your plans though - you could save some time if you switch Koyasan and Himeji/Kobe, and after seeing the latter, go straight on to Hiroshima without the backtracking.
You should know if you don't already that JR trains won't take you all the way to Koyasan. Look into getting the Koyasan World Heritage Ticket.
https://www.japan-guide.com/e/e2358_012.htmlYou can see the best there is at Koyasan in a day, but no matter what, do not miss the Okunoin Cemetery.
https://goo.gl/uAi9oXAnd if you are staying there overnight and want to see the cemetery at night (a really surreal experience), make sure you are staying at a place with no curfew. For Himeji, if you are splitting the day with Kobe, you can see the castle and Kokoen Garden - both are some of Japan's best. See the castle first - last entry will be 4 PM for the castle, and for the garden, at 4:30 PM. Get the combo ticket for both and you can save a little money.
https://goo.gl/65ZtRiFor Hiroshima/Miyajima, 3 days is more than adequate. You could even reallocate a day somewhere else if you wish.
https://goo.gl/fAz4axAnother option is go a bit further and see Iwakuni. The Akiyoshido Cave is also the finest in the country and there aren't many places to see caverns way over your head and an underground river rushing past you.
https://goo.gl/NNQVrVYou could if you wish pull a day off of Kusatsu - one day is enough. There are other sights nearby like Minakami, Karuizawa etc.