They're terms of enderment used to describe feelings for a close friend. There is a difference between a close friend and a lover. I would use the same terms you desribed above with my friend but it does not mean I am in love with her. I say I "miss you" and "I love you" to my bf. I care about you is understood as part of "I love you" . If i started to replace "I love you" with "I care about you" he would think I was loosing interest in our relationship. Think about it.
he says that to woo her and get what he wants he is a guy
If you say "care about you" that means he doesn't want to say "love". Something is making him hesitate.
It means he misses her and cares about her. It is at least on the way to love.
If it was me, yes, if it was most guys? I am not so sure.
It should but only you now that answer!