What is the purpose of having grades?
A professor who I used to work with once gave me an explanation that I found quite useful for understanding the purpose and philosophy of grading. Universities, he said, must always struggle with a tension between two educational goals, illumination and certification.
Illumination means the intellectual development of the student, bringing them a deeper understanding of a subject, its relation to the world, and the deeper issues it may touch on. From the perspective of illumination, marks are intended to be feedback to students that helps them realize weaknesses in their understanding so that they can fix them.
Certification means evaluating a set of skills acquired by the students against an objective standard, to attempt to measure their fitness for certain tasks or professions. From the perspective of certification, marks are intended to be objective judgement of the fitness of the student for carrying out tasks requiring the skills taught in a class.
These two are often in tension with one another because certification pushes teaching toward rote practice and standardized testing and grading, while illumination pushes toward more open-ended exploration and interactive formats which can deliver much more benefit for apt students but are often very subjective. Most classes try to deliver both, to varying degrees of success, though some classes may almost entirely hew to a single side of the balance.
You need to decide what you're after from the classes that you take. From your "pay for service" tone, it sounds like you want the career value that comes from certification. But certification isn't valuable if the standard can be easily negotiated, and so professors have to set a standard and stick by it. Sometimes they are even forced to by regulations. From the perspective of certification, trying to negotiate for a better grade is trying to cheat the system and reduce the value of everybody's grades.
Moreover, one of the "meta-skills" that is always being certified is the ability to figure out what somebody wants from you. If you're out in industry and you deliver the wrong thing because you misinterpreted your client's needs and didn't ask for clarification, it will be difficult to argue that you should be given more partial credit.
If, on the other hand, you're after illumination, then grades are less important to begin with. In that case, it's more important to understand why you got the grade that you got, so that you can improve your understanding of the material. If you want illumination and you aren't getting it, you need to switch courses, majors, or institutions.
Given some of the answers and comments on the question I'm curious, how did this mentality that marks are non-negotiable arise? There seems to be the belief the prof has a totalitarian rule over the students. This doesn't make sense, especially considering how commercialized some schools have become. In any other area of business if a client pays (e.g. a student pays tuition) and is dissatisfied or has a concern about a service, then the company would work with them and either explain or change some part of the contract.
I don't think "non-negotiable" is the right word. Marks can be questioned and even challenged. However such challenges are (outside of the movie Clueless) not negotiations, because that implies a business transaction in which the student is offering something in return.
[Also, professors do not have "totalitarian rule" over the students. We don't have any "rule" over the students. We can only ask them to do certain limited things and they get to decide whether to do them or not. It is of course very common for students to drop or exchange a class because they are not happy with some aspect of how it is being run. This is really the antithesis of totalitarian rule.]
Grading is not a business transaction. You seem to think (or at least be willing to argue) that modern academia is a business transaction in which the student is the client and the instructor is the service provider. Well, there is some truth to that, but it also has severe limitations. (By the way, I have found that most businesses whose services I enlist as a paying client have severe limitations on how they are willing to work with me or (especially) change part of the contract in response to my complaints. The threat of losing my business does something in some cases and very little in others.)
It is worth thinking about what services a university is actually providing, and to whom. If all you wanted from your university was to teach you courses and give you a grade at the end, you could enroll in internet classes at little (or no) cost. Most universities -- especially expensive ones -- are also certifying competence and providing prestige to their graduates. That is why you are paying them the big bucks. This only works if the grades themselves are not negotiable in the sense you mean. At a very elite university there will be considerable resources available for the student and steps taken to try to ensure their success, and the average GPA may be higher than at some other universities. But I taught calculus at Harvard for several years, and every time there were some students who got D's and F's. A B- in calculus at Harvard is a discouraging grade -- literally; it is meant to signal to the student to seriously shape up or not continue studying math -- but it does certify some amount of calculus knowledge. To get an A in calculus at Harvard you must indeed be very good at the subject: Harvard wouldn't be a top American university if it gave top grades to students who had not mastered the material.
Sometimes it helps to make the situation more extreme. If you think that "the student should always be right", perform this thought experiment: I will offer you the opportunity to take the COSATs, a consumer-oriented variant of the SATs. Every student who takes my exam will pass. In fact, every student who gets less than the 50th percentile will have their score reported as "satisfactory". And that's just for basic members. Silver members will be allowed to answer again the questions that they got wrong and will have their exams rescored. Gold members will be offered the same service together with additional instructional materials that will include complete and comprehensive answers to all exam questions. Platinum members get online access to the materials while taking the exam, in a patented "one-click: correct!" format. Of course the COSATs will cost money -- so do the SATs! But actually basic membership is cheaper than the SATs and silver, gold and platinum membership is surprisingly competitively priced. Are you interested?
Grades do many things, and there is no general agreement which of those things is the most important. A few things that grades do are:
Grades motivate students to learn. Without grades, many students would not learn nearly as much, because the mere presence of grades encourages students to study.
Grades tell others whether students have mastered the material of the class (e.g. which students can count a class towards graduation, and which need to re-take the class).
Grades tell others how students compare with each other (e.g. to help determine which students are accepted to competitive graduate programs).
Even when they disagree about why grades are assigned, most professors do want to grade fairly, and will listen to student comments and consider them seriously. But, as I described in another post, the situation may be more complex than is apparent from a student perspective. Moreover, many universities have a system for "grade appeals" through which students can formally dispute grades. So, far from being "non-negotiable", grades are usually subject to review by the professor's superiors at the university.
In any other area of business if a client pays (e.g. a student pays tuition) and is dissatisfied or has a concern about a service, then the company would work with them and either explain or change some part of the contract.
This is not really the case. Imagine two hypothetical scenarios.
I buy an annual membership in a local warehouse club (e.g. Costco or Sam's Club), and then tell them I don't like their bananas, and I want them to buy some other kind. They are likely to ignore me, unless many other people make a similar request. They may well just tell me to go buy bananas somewhere else if I don't like the ones they offer. There are various opinions about whether universities can afford to take a similar position with their students.
I hire a professional opera singer, but then I tell her that I want her to sing pop songs, and by the way can she also lose a little weight and learn how to dance better? She's just as likely to just tell me "no" as she is to work with me to figure out which pop songs I prefer. She has an uncommon skill that is sufficiently in demand to keep her employed. There are various opinions about whether professors are in a similar situation.
The applicability of #1 and #2 to universities can certainly be questioned. These examples are just meant to show that it is not universally true that a paying customer can negotiate freely with the company or person being paid. This goes against the idea that this sort of negotiation should "also" apply at universities.