My old mechanic, a four-bay garage, always had time for emergencies–usually no more than a day or two. I get the feeling that they left a few slots open every day for emergencies.
Since I moved, the new place I go to not only has a week lead time for emergency repairs, they actually take a week off for Christmas. It’s like they don’t actually care that their customers depend on their cars. and just fill up every slot they have as soon as they can, regardless of urgency.
Is always having time available for emergencies a usual practice or an unusual one?
Different places have different norms. You’ll also find shops of any sort that work better and worse with your schedule. Whether or not it works with your budget may be a different story.