First, I can support your idea a little:
An exam or assignment is not always the best way to judge a student's understanding - although it is the easiest and most practical.
An education system might reasonably have some way for a student to demonstrate their clear knowledge and understanding of what is being taught outside the examination or assignment. If there are objective standards that can be proven by the student, this takes away the variability from the examination room or particular assignment and student's/instructor's differing interpretation of the assignment requirements.
It might be, then, that your instructor is willing to consider regrading, if you can provide suitable evidence outside the exam that you understood and can apply the material at a level higher than you did in the exam.
However, I think this is very unlikely to happen in practice:
- There is a serious ethical issue here: all the students should have the same opportunity that you do. You should not receive special treatment just because you personally appeal to the instructor. (The other students may not have the opportunity to personally appeal, or may not realise that it makes a difference). The criteria for grading should be equal and obvious to all students. If an appeal process is possible, then ethically all students must be made aware of the possibility.
[Aside: In your case, if there is an available appeal process, you can follow it with no problem: if you meet the criteria]
Exams and assignments are typically designed to test your understanding in an effective way. They may not be ideal, but usually they are the only good and practical way of testing - there is unlikely to be a good alternative proof you can offer the instructor.
An instructor's time is precious: it is likely they have spent significant time writing an assignment that suitably tests the material, and significant time in marking, and will not be willing to spend more time to listen to this complaint.
It's likely that avoiding "careless mistakes" is part of the test - the grade should depend partly on whether the student understands the material well enough to avoid commonly-made mistakes. So, if you feel you have made careless mistakes, it shows that - probably - you did not understand the material well enough to check and avoid these in a timely manner.
The amount of work you did, or effort you put in, should not really be relevant to the grading. You must show understanding, not effort. (Although, effort should be necessary - it is not what's being tested)
I understand the argument that there is no harm in asking, but I do not expect that this type of request would be well received by the instructor (neither, by your fellow students).
It's plausible that asking for a regrade would harm your professional reputation with the instructor, and your personal reputation amongst fellow students.
I recommend that you consider asking for a retest instead.