The header-title-area.php, unfortunately, cannot be used as a template for changing the code in the child theme. This behavior applies only to page templates and files like loop-course.php etc. which have been marked as template and loaded as such. The child theme file will simply be ignored if the file is not loaded as a template.
The way to modify the header-title-area.php is simple enough still – you just override the methods in the parent file in your child theme functions.php or any other file included by the functions.php of the child theme.
So, if you have a method named function mo_display_top_header_area() {} in your child theme with the extra fields you need, it will override the method defined in the parent theme file. Pls give it a try – it works quite well actually.
This way you don’t have to have the whole file in your child theme – just the methods you need to change would suffice.