What is the point of this doorway "header", and can I simply remove it?
Yes, you can remove that. Its purpose was solely to support drywall and whatever door or trim was installed. It is a header, just not a load-bearing variety. If it was there would be something transferring load from the plate above, and it would be taller.

Modern code aside, you may find that a pre-hung 24" door (should you find one available) slides right in after your rough opening height adjustment. If you can't find one, it's not too difficult to reduce a wider door and jamb.