This is similar to which-func.el, which displays the trailing element of
the breadcrumb in the headerline. Perhaps the breadcrumb feature can be
thought of as an augmentation of which-func.el's functionality. I don't
have a strong opinion there.
I see this feature as an augmentation of which-func.el. The purpose is
to show what is the current defun, but the how and where (mode line or
header line) is a configuration option.
In recent versions of IDEs and text editors another header line UI to
show the current function is getting popular, something like our old
Sticky Function mode. As with this breadcrumb UI, I think its logic
belongs more to which-func-mode.el and shouldn't be coupled to Semantic.