The thing is, we all have different ideas about parenting and what we think is OK. So here's a way to check whether you're BU.
Do a reverse. Your partner has paid for you, him and your joint DS to go away for Christmas. You have your own, older DS from a past relationship who has said that he'll go to his nan/dad/other relative for Christmas. Older DS normally lives with you. A couple of months before you're due to go, he says he doesn't want to go to the relative and wants to come on the family holiday with you. Your partner says no, he can't come, he has to stay at home. He'll pay for your joint child to come but he won't pay for your own DS and doesn't want him there. Your partner thinks your son is a hassle and wants a quiet break without him for a change. Not a child-free break obviously, as you'll still have your own joint DS. He just doesn't want your DS, his SDS to come. Your son will have to stay at home on his own but he's upset because he feels excluded from your family holiday. Are you comfortable with your partner excluding your son from your family break and going away, leaving him on his own? Are you comfortable with your partner making it clear what a hassle he finds your son to be and wanting to get away from him (while still taking a half-sibling)?
If you can say, hand on heart, that you'd be OK with that decision, then fair enough. But the thing about being a step parent is that you can't treat your stepchildren worse than you'd accept your own children being treated. Where that line lies is up to you as a parent, but it needs to be consistent for ALL your children, bio, step or otherwise.
In all honesty though, reading your update you sound incredibly hostile towards your stepson, and resentful of the responsibilities it places on you. He may be 16 now but if he senses your negativity towards him, it's going to hurt and have repercussions.
You say you're going to have a rethink. It's OK if being a step-parent isn't for you - it will get easier as he becomes an adult but he's always going to be part of your lives, and you'll always need to consider him to a greater or less extent. Whether that's what you want is up to you. But you can't treat him like an annoying hanger-on for the rest of his life.