That is not "arithmetic", that is computation, just a part of arithmetic. And your examples are sport, which is fine. Arithmetic also includes recognizing the arithmetic in various scenarios and contexts. Knowing when to add or multiply. Recognizing basic and fundamental patterns like the commutative and distributive properties. Understanding the nature of physical quantities and the relationship of the "value" and the "unit" to them. Knowing how to write and say numbers and use the 13 characters (0-9 period comma minus). Understanding place value. Knowing fractions, both vulgar and decimal, and recognizing that they are numbers and have their place on the number line.
You really don't appreciate all of this until you actually teach a young student all of this. We take it for granted. Also, we don't remember the experience of learning all of this because it occurs before we know enough to put it in perspective. We might remember the setting, but not the experience. We remember what it was like to learn algebra but not what it was like to learn our ABCs or to count or our first exposure to adding and subtracting. By the time we start to have genuine memories of learning late in primary school, or sometimes not till middle school, after the basics have been covered.
Besides its practical importance, arithmetic is about becoming familiar with numbers, especially real numbers. What could be more important to higher math? I didn't say "also important" I said "more important"?
When you actually teach children, what to teach comes pretty easy, but I suppose that depends on what kind of teacher you are. I start a lesson with my son and I know right away if I am missing something. When you have the teaching bug you are able to be both the student and the teacher at the same time. This becomes more and more difficult when the class is larger and more diverse in level. Obviously, it breaks down completely when the class has a 100+ students, most of them unprepared and uninterested, like we are seeing in these large required college introductory classes.