U.S. grading system: Why discrete grades?
I am in favor of using the US discrete grade system. Some thoughts I will address are as follows:

As the OP mentions, in the US, the ABC grade system is relatively standardized and well understood by a large variety of institutions. It makes comparisons among students from a variety of schools more compatible (although, yes, not perfectly comparable). 

Beyond this, I feel that the ABC system delivers the appropriate amount of granularity. After all, you have to truncate grade percentages somewhere. In some sense every system is discrete. 

One method that I have used for my students in math classes in the US is a clustering system guided by the grade cutoffs. The percentages are guarantees. After that, I cluster students into grade groups, with students who barely missed the cutoff still usually getting adjusted up. For example, let's say that we have the following percentages:


  93.23%, 93.17%, 92.88%, 92.81%, 92.08%


I would give the first four students A's and the final student an A- under normal circumstances. (It is usually more nuanced than that). I feel that the grades I gave my students usually very accurately described their proficiency relative to the stated outcomes of the course.  This was also vastly true when I was a student. In fact there were times where I would prefer to take my A grade than my raw percentage. An 'A' sounds much better than "I got 83% raw, but I was the second highest grade in the course."

In my opinion, the ABC system actually is a stronger system than a raw percentage system. Is a 95% in Dr. Haskin's Calculus I class better than a 97% in Dr. Bodrel's Calculus I course? Both would result in an A grade. Or how would a 97% in Calculus I compare to a 97% in Floral Design? By using the ABC system, we effectively run a sort of "low-pass" filter on the grades. In my opinion this actually allows for a fairer comparison between students. 

Let me also add that having a raw percentage system would be awful from the perspective of an instructor. Students already anguish about their grades. Now picture if they had to anguish about not just getting an A, but getting 100%. You would end up having students with 98% in the course worrying about getting 99%. It just becomes excessively fiddly. As a former student myself, I would actually hate a raw system. I was grateful for being able to get an A with 94% in a class without having to worry about pushing it up to a 95%. 

I will close by saying that school and grades are not always fair. Such is life. No matter what system you use, there will be drawbacks from one perspective or another. There is no system that is perfect. 



Addendum: Several comments have mentioned thoughts on being graded relative to others. Note first that my proposed method never moves a student down a grade, only up. Several comments seem to overlook that. 


  @BartoszKP: "You are grading not 'their proficiency relative to the stated outcomes,' but 'their proficiency relative to the stated outcomes and relative to other students.' Did other students doing well make this student suddenly know less?" 


How could raising a grade indicate a student "knew less"? It seems that raising a grade would actually indicate a student knew more. 

I guess I could grade students solely on their raw scores, 100ths of a percent be damned. Then students would be graded with less relativity to their peers. But wasn't such a "zero-tolerance of rounding up" system one of the named drawbacks of the ABC system? It seems like we are trying to play both ends of the flute here. I can see that it would be patently unfair if I rounded students down a grade. I must say though, I have never had a student complain about being graded "relative" to their peers when I rounded their grade slightly up. 

Secondly, to continue the thoughts on being graded relative to one's peers, we need to keep in mind that it is next to impossible to provide students from current semesters the identical exam experience as students from previous semesters encountered. By adjusting scores within semester, this can help control for slight variations in exam difficulty from semester to semester. Note that I do not always have complete control over exams, since they often must be written on teams with other professors. All told, which system seems more "fair," a system that does not control at all for variability in exam writing styles or a system that attempts to account for that?