“Fix it tickets” are not a national thing. Some places do it, some places don’t. It’s just easier to say with a general meaning like Kleenex versus tissue paper.
Many places just issue you a ticket. End of story. Once the ticket is issued you are on notice that your vehicle is not considered legally roadworthy. You are legally supposed to park it and not drive on public roads until fixed. Tow it to the shop if you have to.
Obviously this is onerous and stupid. But it is the law. If you have a minor issue, 99% of the time the pig will let you go and get it fixed like you said. But they don’t have to. “Officer discretion” is a fancy way of saying applying laws on a whim. Picking and choosing when to enforce laws and against whom.
In some countries it is much more clear cut. If you need to fix something the police escorts you to a staging area, like a big parking lot and that’s where your car sits until it’s fixed. Once fixed the popo there check it before you drive out. People in the lot changing bald tires, burned bulbs, fastening new side mirrors, etc.