The earned not won distinction is almost a cliche. On the one hand, I can see how "won" might be construed as "acquired by chance" (like I "won" the stuffed animal at state fair or I "won" the lottery). But on the other hand, "won" could mean "she strove", "she rose to the occasion", "she fought for it" or "her efforts exceeded the efforts of others" as opposed to "she 'earned' her pay" or "she 'earned' her stripes" etc. I know what you mean though, Elizabeth probably thinks the soldier was lucky.
No. Saying she won the Silver Star makes it sound like a contest. It also implies that all those who didn't get a medal are losers. It is not a cliche; it is an important distinction.