I'm in the "nobody comes first, all members of the family are equally important and needs are accessed in each individual case" camp. There should be no need to say one person always comes first, in a family where everybody is considerate and considered.
this is so healthy.
I can think of dozens of situations where my kids had to take a back seat, which doesn't mean that they are not my priority, or that they are not important, just that in life they do not always come first.
eg dc1 went to their first day at school with Granny not Mum and I don;t even have a photo. Why? dc3 was in hospital. So dc 2's needs had to take second place to dc3.
eg my mum was in hospital, and I had to drive my dad to visiting time, right at kids dinner/bedtime when they were about aged 5-10.
For that few weeks, Daddy not Mummy did most of the dinner and bedtimes, My mum took priority.
eg Last weekend my dh and I went away for a much needed break. My 15 year old was fine (looked after by older sibling) but predicatably had a melt down on Monday, as she always does if routine etc is disrupted. (suspected autism) Should we have stayed at home? No, we handled the melt down wiht love and hugs and acknowledged that she has had been amazing making it possible for us to go.
A new partner should never ever push kids out of their home or emotionally out of their family, but do kids ALWAYS come first? No.