Yes, you can define addition in terms of subtraction if you want to. But suppose I have a set $S$ which is non-empty and closed under subtraction as I would recognise it. Then:
- $S$ contains $0$, since it's just $x-x$ for some $x\in S$
- Hence, for all $x\in S$, $-x\in S$, since $-x = 0-x$.
- Hence, for all $x,y\in S$, $x+y\in S$, since $x+y=x-(-y)$.
Hence $S$ is a group under addition. The converse doesn't hold: there are structures, like the natural numbers, closed under $+$ but where $-$ doesn't make sense.
From this I conclude that $+$ is a strictly more general operation than $-$, hence more often applicable, and therefore (in addition to it just generally being "nicer" in the sense of associativity and commutativity) more worthy of study and consideration. Wherever you find subtraction, it's because there's really some addition around, whereas addition need not come with a corresponding subtraction operation.
And one more thing: ask yourself what subtraction means without referring to addition. The simplest way to describe the concept of subtraction, rather than the implementation, is certainly to refer to addition, which itself is described in terms of the successor operation or something similar (the cardinality of a disjoint union, perhaps?)
: Pun... sorta intended. Don't judge me.