When you had your animal put down, where you in the room when they did it? Did you bring you animal back home for a burial, have it cremated, etc? Did you do it when the office opened, or at closing time?
Yes, we have always been present, and we've done this several times. The first time was the hardest because we didn't know what to expect.
If there's not an emergency and we can choose the time, we do it earlier in the day, so that we have daylight to bury the animal on our land, close to our other animals who are buried there. We have a lot of acreage in a rural area. (If possible to prepare, we pre-dig the hole, sometimes over a period of a few days.) Also we want to be able to really take our time at the vets' - it can take us a few hours - so wouldn't choose to do this near closing time.
We usually do the euthanasia outdoors at the vets' backyard, where our animal might enjoy the grass and sunshine and the scents of other animals, rather than in a sterile examining room. We bring towels or blankets and a crate bottom for transport (and a pooper scooper if we remember, just to mention the practical aspect of this, tho of course your vet will have one).
Also, we always opt to get 2 injections rather than one. The first is a tranquilizer and induces a more gradual peaceful sleep. (This is a few extra dollars for the tranquilizer shot but worth it, primarily because it's easier emotionally for us. Also- usually the euthanasia shot works immediately, but occasionally it doesn't - so this way we're more sure they can't ever feel any discomfort.) The vet doesn't give even this first shot until we're ready, so sometimes we spend a while out there on the lawn first, if our animal is comfortable and alert enough to be interested in the environment.
Between the first and second injection, we stay with our animal and hold them and pet them, and talk to them, and cry and pray or whatever. The animal falls asleep pretty slowly so they can hear us for awhile, and then they are still breathing and sleeping while we sit with them. The vet gives us as much time as we need, an hour if we want, checking in with us a few times if necessary. Sometimes it will be the first time the animal will have seemed so relaxed in recent memory, assuming they've had discomfort or pain, and this is very relieving part of the transition. The vet never pressures us to rush.
Then after the second shot, we can also stay right there as long as we want, until we are ready to go home.
One regret was our most recent loss: even knowing things were pretty hopeless for our dog who was in a coma, I opted for an IV drip, hoping for the .0001% chance that he would recover. He didn't of course, and we weren't there when he died, which happened in the evening when we couldn't bring him home til the next day. That was much more difficult for me.