I went on a spending spree and bought a ton of random geldings to build a show string. (not counting yearlings and two year olds) but the ones that are 4 and older, there are several who have never gone green in the time that I have owned them. Will they ever?

I've taken a quick glance at your 100 geldings barn. I found several who haven't been placing well because they haven't been ready to show then. A PPP+ gelding needs at least 5 weeks of training after they'd leveled up before they're ready to show again. If they don't have old show reports because they have never been shown before, take a look at their training bars. In general they need twice the number of bars to show in a certain level. So e.g. a lv 2 gelding to show for example in Dressage needs overall 12 bars in mv, int and str. It doens't matter how the bars are distributed.

Then there are sometimes horses not placing well because they're low in hrt. A show horse low in hrt is more likely to get eliminated in CC and SJ so I usually cull them.