Having more say in the horse's care was important for me (I'm definitely particular about my horse's nutrition and hoof care), but the big thing that really got me motivated to buy instead of leasing was that I kept starting over again with lease horses.
The first horse I leased was older (mid-20's or so) and got retired after I had ridden her for a year or two, so I moved on to a new horse. I was definitely a beginner when I started riding her, and she was great to learn on. I don't think I would have been able to handle my 2nd lease horse if I had tried starting with him!
He was younger, but was suspected to have navicular disease and the owner decided to try a barefoot trimmer with him to see if he could be made more comfortable. After several months of not being able to ride him consistently I decided it was time to move on to another horse. It turned out later that whatever his issue was, it was confirmed by x-rays that it wasn't navicular, but to this day he's still lame some days and not-so-lame others. I did click with him very well and if I ever get my own horse property I'd love to buy him as a companion for my horse.
The third (and final) horse I leased was fine health-wise, but had a bit of a bucking problem I was pretty glad when his lease was up.
I lucked out that I had access to ride my leased horses pretty much whenever I wanted (even though the first 2 were technically half leases). None of them really had solid dressage foundations prior to my riding them, so I always ended up starting at the basics again and could never quite make it to 1st level before switching to a new horse for one reason or the other. The horse I ended up buying had some informal dressage training, but hadn't ever competed. I started him at intro level last year, and we just rode in our first recognized show at 1st level a couple months ago