So what exactly do half pads/wither pads do? Obviously they have some benefit but I’ve read a lot of different reasons for using them. So what is the real reason?
I imagine there are a lot of different reasons for using them! 
To improve the fit of a saddle, or because your horse seems happier with a little extra cushioning? The pads that take shims can be used to fill in horses with the kind of wither or back atrophy that no saddle panels could adequately fill
I’ve used them to fill and resolve minor(!) issues with saddle fit, and seen them used for (minor) saddle adjustments on young/out of shape horses who are still filling out, to avoid multiple saddles in a short period of time. They can help out when you can afford an almost perfect fitting saddle, but can’t afford a custom perfect fitting one.
Ogilvy/Thinlines supposedly also absorb some of the concussion of the rider, so take that with a grain of salt.
I used one to bump my saddle off of my horses withers. She was underweight when I first got her so every saddle I put on her either sat on her withers or pinched her shoulders. So until she filled out I used one of these or a second pad to make up for where the saddle didn’t fit.
I use a half pad for the cushioning but no added bulk under my leg. I do not use it on top of a baby pad or square pad. I use just the half pad. I use a really lambswool so I feel that it is cooler and breathes better in the heat of the summer.
I used to use a shimmable half pad before I got my custom saddle. The shimmable pad helped make up for a saddle that did not fit him well. He has hollows and shark fins for withers.
I used one when I learned to ride to help the poor pony carrying the sack of potatoes (me)
