Assignment extensions for no apparent reasons (USA)
The short answer is that opinions on granting extensions vary between institutions and even departments. Some universities view extensions as a way to support student learning. They argue that they want students to be able to get the most out of each assignment and so when the student completes the work is less important than that they do eventually complete it and are able to focus their full attention on it.

Other universities hold the same perspective that you do, that learning to manage deadlines is an important piece of being a student, and that allowing for extensions whenever asked is doing a disservice to students.

The chair of your department or even the other faculty should be able to give you insight on expectations that you should follow and on what is customary. I strongly recommend that you talk to them, since they know your students.

Another option is to pre-emptively institute a limited extension system for all students. The mathematics and computer science courses at my undergraduate college used this tactic to great success. On the first day of the semester, the professor told the students that each of them had a certain number of "late days" (usually 4 for a class with weekly assignments), which would grant a student 24 hours of automatic extension. These could be used at any time on any assignment except for group work or exams. Students did not need to provide a reason for using a late day, although they did need to inform the professor/a TA that they were. There was usually a form online for students to fill out. Students could use them as they wanted- all at once or one at a time. Some professors gave small amounts of extra credit for having late days remaining at the end of the semester, and some allowed students to earn extra late days throughout the semester by doing extra credit work, handing in assignments significantly early, or going to relevant talks, etc.

The benefit of a system like late days is that most students will feel less stressed and more in control of their own time. It thus enables you, as the professor, to hold a quite strict "no other extensions unless absolute emergency" policy. It was well understood that if a student used up all of their late days early on and then got minorly sick or had a job interview or had something else come up, that was just too bad. They would turn in what they had. Further extensions were typically only granted if the student or a family member was in the hospital or if they had formal accommodations through the college itself.