From my perspective of working at a clinic, it gets a little bit annoying that we squeeze everybody in because it definitely pushes us behind schedule... which, would be okay, except the OTHER clients who have appointments that we aren't getting to on time, never seem to understand. Their vet appointment is just one appointment of many, they have other places to be and they get downright ticked off if they have to wait. I always hope they can appreciate the fact that this is what we would be doing for THEIR pet if it were needed, but no one ever seems to get that!
We have emergency spots that aren't supposed to be filled until the day of, we offer a "sit and wait" option ("I literally have no appointment time to offer you, but you can come on up and we will see you when we can"), or you can drop your pet off. For TRUE emergencies, which we rarely see (ie hit by car, actively seizuring, respiratory distress or some such), we'll basically drop everything and get to work on that animal.
I think we are suuuuuuper accommodating when it comes to getting people in, even new clients get squeezed in a lot.