What we see in College is that

kids in America don't have enough training in proofs. We spend too much time on it.

Proofs are everywhere. When a student designs the steps in solving a problem, when a teacher introduces a new skill or theorem, or when a parent explains to his kids the solution of a problem, it is mathematical proofs that they are using. Giving the same problem to the same class, why some kids have the solution immediately while others struggle? Because for the first group, they understand the reason, i.e., the proofs. Sometimes you just take it for granted. But whenever you apply a mathematical theorem, that is a proof--you are convinced that a particular method works in this situation.

BTW, looking for alternative proofs is never for proof's sake. We are trying to understand the subject from different angles. That can not come from nowhere. Usually we have to speculate on things already known to get new ideas, and then apply them to more advanced problems. In other words, we are trying to understand the elephant as a whole.