Do you have a public page for this so people can see it in action? It simply feels to me that you have taken the element out of the flow of the page and that the header was being centered because of some parent element… that or the element itself is just not being set to be the full width of the page because it was full width of a parent container (that now doesn’t apply).
It is hard to conclude that unless we can see and test some changes on the problem page.
But if the child is what you are making fixed, it will pull that element out of the flow of the page and no longer controlled by .wrapper. But again I don’t know if we can’t see an example.
Then the fixed element would need to be 1190px wide because fixed elements are shrunk to fit. you would need left:0 ;right:0; width:100%; max-width:1190px and margin:auto.
I would stick with position:sticky and avoid those issues
If you are going to use position:sticky then do you still need to add a class with JS?
Distraction-free: I am planning to give a distraction-free fixed/sticky menu. when the user is scrolling down the menu should not appear, when he scrolls up it should appear.
It depends on the layout of course because you may want some inner elements to be full width which will mean having them outside of the wrap would be more suitable.
Generally though if you have a centred layout then having a wrap means you only have to specify the dimensions once as all inner elements will be controlled by that wrap. This makes it easier for media queries as you only need to address one element. In your second element you would have to specify the width and margins for every direct child of the body and do the same when you change them in the media queries.
Some people like to use the body element as a wrap which is ok for simple designs but I often find it more cumbersome that way rather than having a wrap and leaving the body well alone (older browsers had many bugs when the body was sized or had margins).
In the end its up to the developer and how they feel things will work best. I prefer to use a wrap because that’s what works for me the best.