First, my experience was in the eventing world, not hunter/jumper. and involved “free leases” (take over all the expenses, but no “lease fee”), and were not associated with a particular barn.
After I lost my own horse to septic arthritis I wanted to ride again, but I wasn’t ready to start horse shopping. I leased a (14h 3") Anglo x Arab x Trak from my vet. He was rehabbing from a suspensory injury, and needed slow consistent work. Once he was fully sound I competed him for several years (at Training) while I did start horse shopping. When I sent him back the owner had a sound horse, with a competitive record he hadn’t had before. We both benefitted.
I also (somewhat overlapping) leased two green horses from the same owner, (one with 10 rides under saddle, one with 2) . Again, she got back horse that were better than they started, and I got the advantage of learning a lot of lessons about training a green horse.
The advantages of leasing are:
If you need one horse now, but know you are going to need a different horse in a couple of years, you can just send the lease horseback, you don’t have to go through the sales process.
If the horse loses value wile you are leasing (age, injury, behavior) you don’t take the loss when selling.
If the horse doesn’t work for you, you can just send it back, you don’t need to go through the sales process.
The disadvantages of leasing:
If you get really attached, it can be hard to let the horse go back to the owner
The owner may decide to end the lease even if you want to continue.
There is the potential of disagreement between the leaser and the owner about the care, treatment, and activities of the horse
Some people say it is a disadvantage that you don’t get to profit from an increase in the horse’ value(that you contributed to) , but, based on my later experience buying green horses and selling them a few years later when they were trained, this is not a significant issue.
ETA that my experience is that lease horses largely pass by word of mouth.