What I find confusing is how you imply that "going to the bank" is an OK reason to leave work, but taking a sick pet to the vet is weird? What country do you work in?
It's a well preserved secret, nobody knows how they did it.
It is clearly magic or sth like that
Spoiler: they do like everybody else's does, they ask for some free time for family.
You have the right to it, even in shitty US
I've heard that in the advanced nation of Germany lots of useful businesses aren't open when ordinary workers have time to go visit them, such as banks and electronics stores and grocery stores. If you work an extra hour for some reason you can't pick up food except at a stall in the train station?
Unless it's structural. Take your kids to school every morning and come in late? That has to be talked through properly. And if you go home early every week because your fashion items are going to be delivered, that's not okay.
Other people take vacation hours for this kind of stuff. (we get 25 days a year). Others will work an hour unpaid overtime here and there, and compensate by going home or getting in early at other times.
In general it's just not an issue really, you just notify upfront and not be there, and make sure not to do it more than twice or so a month.
It really helps that working from home 1 day a week is becoming more of a thing though. You're more flexible that way.
I ask my boss and he says ok I hope your dog is feeling better. 8-5 Megacorp job.