Yes, I've had those moments with pets I've put down. A cat we had for 12 years, my first dog when he was 12, another when he was five after escaping and drinking anti-freeze somewhere...come immediately to mind.
I was also there and holding on when my Father died (I was 19) and when my Mother died 18 years later. The elder generations had that experience in their lives, and while I don't wish it on any one...it really does help you understand death.
Before, I didn't fear death because I was young and therefor immortal , but after - even as I advance in years - I don't fear death because it IS the track for all of us. While I don't wish to hasten it, I do hope it will come on swift wing when it strikes (I've given up on glory ).
But even if it comes slowly, like the increasing time it takes me to warm-up on a winter day, I'll just say hello and pass the time of day as it will gladly stop for me.