I was there with the two I’ve had to put down over the years. My vet handles getting them down ( and I don’t watch that part since it can go wrong). But it was important for me to be there at the end, looking them in the eye, stroking their face. I think it’s important that someone familiar is there. Maybe that’s overly sentimental, but I felt that was my responsibility. It can be incredibly hard ( especially the young horse I had to put down because it felt so unfair and that he should have had more time). But I don’t regret being there at all. They do so much for us, I think it’s only fitting we try to be there for them.
As others have said, I don’t stay for the removal. The cremation company handled that, and they’re pretty adamant that owners shouldn’t stay for it.