When I wrote my response, I didn't realize your group size. I think with a group size of 6 or more you are going to have to live with some less than ideal camps, etc. For instance a camp like MOL only has 2 sites. It's very unlikely that you will have the option of getting both of those sites especially in the walk-up scenario. Other popular sites like FIF and GRN would be tough as well.
It wouldn't make sense to hike all the way up to the STO and then hike down and over to FIF. The trail up to the STO from the west is not so great and usually pretty hot. It doesn't afford many views either.How about this one.... GSE - KOO - STO - FIF - GRN - exit @ MANY is that one of the most popular ? Isn't hat supposed to be one of the prettiest routes....?
I think the route you were quoted is OK, probably better than 4 out of 10. An improvement would be to add a camp between ELH and GLH. If I were going to try and improve this route I would do something like (SLI - GAB - ELH - COS - GLH - KOO - GSE). Adding COS will shorten your 4th day and allow you time to relax at the lake. It would also give you extra time the next day.