I think it entirely depends on the parent.  To use your example of skiing and a person who would like you to accompany them, do they spend time getting you used to the equipment, play in the snow with you, start on the "bunny slopes", and get some mulled wine afterward while telling you how well you did?  Or do they take you to the tallest mountain, say "go downhill, real fast, and try not to fall" before taking off themselves and leaving you stranded up there with no way to get down except how you can manage in the cold and wind with weird sticks strapped to your feet?

If a parent actually wants you to be a part of their life with the child and puts in the work then it is reasonable to make the effort in return for a happy relationship.  But if they do not and the child is just thrust at you as something that should magically happen through sheer force of will then you are not obligated to do anything in particular.  They dropped the ball and it is on them.  Sadly that happens a lot.  Many parents have this weird belief that, because they think their kid is the best thing that happened to the world, everyone else will feel the same way.  No one else does.

Courtesy, yes, but I have no real interest in my SDs.  They have forfeited that through their vile behaviour to me for many years.   Since 2013 DH has had much better boundaries regarding this, but the damage cannot be undone. 

I think everyone should get basic courtesy and respect, at least until they demonstrate to me that they don't deserve it.  

My SS likes me, but if he didn't, and he was rude to me, I wouldn't go out of my way to try to have a relationship with him, but I would still treat him respectfully. 

You can't make yourself care about something just because DH does. My DH loves gaming, it makes me dizzy. I don't need to force myself through dizziness to make him happy.

Nope.  DH can care for his prior relationship child all he wants. But... your care for that kid should be dependent on that kid's behavior and performance.

An accident of procreation does not make someone important to an unrelated person just because a spouse brings that accident to a new relationship. That kid earns their place in the new blended marriage dynamic..... or not.

Entirely separate individuals. Whenever someone tells me their kids are "a part of them" I ask them if they view themselves as a part of their parents instead of an entirely separate person. The answer has always been no.

I feel the same about skid. I have trouble seeing her as DHs at all because shes a clone of BM. People comment so often on how strange it is that shes nothing like DH looks or personality wise that it makes it even harder because the idea that shes mini BM gets driven home even more. Even strangers like stores and stuff dont realize that she is with us, idk if it's because she looks like shes almost a different ethnicity or if its everyone's body language toward eachother or what.

Also, BM used the "she will always be a part of me" as part of her control tactics toward DH, so that phrase feels very manipulative to me. 

BM and DH’s parents were like this in the beginning. I think my in-laws still feel this way.

It will never happen, as I am beyond indifferent when it comes to skid. 

When I read your title all I could picture was this weird child shaped appendage sticking wrongly off of your DH. So I guess that's my view when somebody thinks their child is an extension of them, lol.

Narcissists view their children as extensions of themselves... 

In my case, I spend 4 days a month with SSs and BM has created an environment where we are not allowed to ask them what goes on at her house and where she sends nasty messages after nearly every visitation complaining about something. That ensures that SSs are so removed from me that I will never feel love for them. They are houseguests who arrive EOWE. Sometimes they are pleasant houseguests, other times, they are unpleasant houseguests, but they are always houseguests. I can't be myself around them - I have to watch what I say and do - so they would never be an extension of my DH. 

Also, following that logic, everyone should love their in-laws as their own, because wouldn't parents also be an extension of a person? I know children need different concessions while they are children, but children eventually become adults and children also have difficult personalities and can be jerks, even though they are children. I often equate the SKid relationship to an in-law relationship. Especially when you are married to the NCP and see them infrequently. 

I treat everyone with respect and decency, it doesn't matter if they are my H's kids or not. If I'm not respected and treated cordially in return, all bets are off. It doesn't concern me if he would feel they are an extension of him or not. I set the tone for what I allow in my circle and treat those accordingly.

I think it depends on each individual step-family and their own history. With some, there's too much hurt and missteps to really extend that type of interest and grace to the SKIDs. Surface level respect and tolerance in those cases are probably for the best.

For my situation, we're not in a HCBM situation and my SKID is generally likeable (and young), so I do show interest because it helps foster our relationship and it's something I want to do to continue to grow that relationship. That might not be the best avenue for others, but for us-- for now--- that is what works

Failed parents raise children to be extensions of themselves.

Successful parents raise their children to be independent viable adults of character and standing in society.

Not a complex concept in the least.