Partially, I think you have this view because you don't know the motivation are behind some these concepts. This is not a bad thing. Yes, sometimes it is more difficult to learn or even care about something, if it seems like it was just invented because someone could do it. The fact that you may not know why something was invented may even be one of the benefits of the modern mathematics approach. Often the author invents some concepts that comes naturally from solving some problems. This concept may have become quite useful, and mathematician have since abstracted to the essential properties, perhaps to the extent no longer recognize the original. However, with much of the "distraction" removed, properties became more evident, theorems were easier to prove, and results became more broadly applicable.
A nice example may be the abstract definition of a topological space. Upon the first encounter with this concept, one may feel that this is some nonsense set-theoretic fabrication. However, this led to other abstract ideas such as connectedness and topological definitions of compactness which has since given easier proofs or some sort of deeper understand of familiar results such as the intermediate value theorems or the extreme value theorems from calculus.
Another branch may be Number Theory. You should take a look at the sort of concepts invented in attempt to solve very simple equations. However, these results have since resulted in concepts like Dedekind Domains, Dirichlet Unit Theorem, Class number, etc that apparently are useful for other aspect of number theory.
So because the fact you feel the unnaturalness of some aspect of mathematics may very well be the result of method of mathematics that has since proven quite useful in solving problems. However, before you judge something to be invented, you should look at its history. However, personally, I would like to do mathematics without having to know all the history, the antiquated methods, the inefficiencies, and the haphazardness of past research. This unnaturalness may just be the fact that these results are just too polished that you don't see the sweat and groans of the mathematicians who had to produce them.
As for your questions of logical foundation ... Do you even consider logic as one of these branches of mathematics that are not "games"? A lot of people don't. Inherently logic is abstract and philosophical, especially foundations. You are trying to create a framework that can be used to formulate anything you want to say. Some aspect of this question is clearly philosophical. As to the existing foundations, I think ZFC set theory is already quite intuitive. Everything is a set. There are weaker logical systems such as second order arithmetics. However usually they give up logical expressiveness (in general harder to use in practice).
I think the existing language of mathematics is sufficient for even some of the applied areas. Physicist often do use concepts studied in mathematics and even phrase them in those mathematical terms. Perhaps set theory and other aspects of mathematics may appear abstract to engineers and others whose line of work requires only the ability to describe numbers, geometries, etc ... whereas the foundations set theorist wants to express everything mathematically.