I'm on the fence of schedule vs natural.
My childhood pet, which was still my parents pet when I was an adult, was in bad shape. They couldn't face the scheduling of putting down. Then one night on a routine pee break outside, he was unsupervised and walked out to the lake and fell in. Parents searched all night for him, finding his body in the lake the next am. Awful way to lose your loved one.
My pet more recently, after months of declining quality, we decided on a Friday to bring her in. Thursday night, she died in my arms unexpectedly.
One viewpoint, they are at home in a familiar place with you, but you never know if something unintended will happen like to my parents, and I don't know if my little one was in extreme pain or not during this last moments.
Other way, they are in strange place, strange people, but you know it's coming and can prepare for that last little bit of life and it a peaceful ending.