Not sure what the veterinary hospital/clinic situation is like near you, but before I had to have my mare put down, I had never thought about euthanasia in that setting. If anything, I would have expected it to be perhaps too clinical and not allow me to properly say goodbye.
As it happened, after months of trying to get a lameness work up appt at the vet hospital because my vet was unable to fully diagnose at the farm, we ended up at the hospital for an emergency visit due to an infection (in the knee, likely from compensating while getting up or down), and while there, they were able to determine that the lameness included severe laminitis that presented unusually (in one hind foot) and only her stoicism had been allowing her to continue on it. It was too far gone to try saving her without putting her through a very long uncomfortable period for very little potential gain, so I opted to put her down at the hospital.
The experience was so much more pleasant than any on-farm euthanasia I have ever assisted with.
The staff were lovely, and brought me treats and grain to feed to her while I spent as much time with her as I needed before we were ready. They used an anaesthesia room to lower her to the ground gently, gave me as much time sitting down with her as I needed before the final injection, and gave as much time as I needed with her body afterwards. They even took care of braiding her tail and cutting it for me when I mentioned I wanted to keep it.
I was given the choice of paying to have her cremated alone, or paying just a very low fee to have her cremated with other animals, which is what I went with (I wasn’t interested in her ashes).
I also signed off on them being able to use her body for teaching, which allowed me to feel that something positive could come from her death.
I was able to leave her lying peacefully in the anaesthetic room and not have to worry about or watch anything that came after.
The only negative was a week later when I received a call from the hospital saying they had an appointment available for her work up; I wish their system had flagged that and not put me in the position of answering that call while my emotions were very raw.
All that to say that if the situation presents itself in such a way that a hospital euthanasia is an option, don’t assume that trailering your horse back home to do it on the farm would necessarily be the more pleasant ending.