I too have had a problem with grade-grubbing in my classes. About a year or so ago, I created a "grade appeal" policy. Before I get into that, I should probably mention that I do not provide any feedback on final assignments. Instead, I provide feedforward. Students email me their assignments one week before the assignment is due and I provide feedforward and a pencil grade so that they can benefit from the comments before actually submitting their work for a grade.
My "grade appeal" policy states that if they feel that their grade does not represent the quality of their work, they have to write me a letter explaining why, and how, they feel their work has meet the objectives of the assignments. They have to use the assignment guide, my grading rubric, and examples from their work when making their argument. I will then schedule a one-on-one virtual conference to discuss the assignment and their appeal. I've had students contact me about their grades, without the appeal, and I respond by saying that I would be happy to discuss the grades with them after they've submitted the appeal. They don't submit the appeal. I've only had two students submit a grade appeal in the last year and my evaluations have not been negatively affected. I've found that grade grubbers are pretty passive students.
I know my policy may sound harsh, and perhaps even unfair, but I in the past several years, I've had a real issue with grade grubbers (and issues related to entitlement, in general) in all my courses, particularly at the end of the semester. I did check with my supervisor and found that this policy was perfectly fine because I allow students the opportunity to discuss their grades with me.
I like the billion points idea, too, though.
I always offer them the opportunity to submit an assignment early for feedback" (I LOVE the term feedforward, btw), but had never considered it as a grading policy. Instead I use 2 things: 1) a grade appeal policy similar to what you describe and 2) "I appreciate your concern, but this was mentioned in the assignment description and you failed to take advantage of my offer to provide feedback before the due date. Had you done that, you might not have lost these points."
Shuts up the grade grubbers fast, but leaves the door open to legitimate errors or concerns.