I just tested now and the header is still coming up when you "triple click" on the paragraph.
I really don't like when things are enforced. I want the header there when I want it and I hate it there when I don't want it. Teams even created a feature to forward messages to others which comes nicely showing who sent it.
MS should create a simple flag "Copy header when copying entire content of a message" and if I mostly need it, I mark it, and if I mostly don't need it, I uncheck it. It would make our day by day less annoying to have to avoid this "enforced" behavior that a single person in MS decided "it's mostly needed" based his very narrow / ignorant point of view and his sole decision affects millions of people on daily basis since several years already.
As an efficient work-around, once you triple click on the message, hold Shift and hit keyboard left once, it will unselect the hidden content at the end of the visible text that represents the header.