I can understand how having a headlamp may seem to be 'extra' to the basics of a first aid kit. For a stripped down kit with the bare essentials (that fit your specific needs) a headlamp may not be a critical item.

However, working and training with cops, prison/jail staff, emergency management and other first responder types I usually see they have a flashlight - the really experienced and seasoned people also have a headlamp to supplement the flashlight. I worked rural areas along with modern state-of-the art detention facilities and having a headlamp to use while working a scene such as an accident, assault or power loss having the headlamp was a literal life saver. I fits in a pocket or as part of a pouch very easily.

I currently have several of them and routinely carry one with me. There are several very small, light weight but bright ones available that are no bigger than a box of tic-tac mints.
Example of one I have: https://www.amazon.com/Black-Diamond-Ion-Headlamp-Size/dp/B01M0WXGM4/ref=dp_ob_title_sports

YMMV but having a headlamp in a kit that is anything beyond the bare basics can be invaluable. They don't cost much and can be very small - I haven't had an actual emergency use for one in several years now but if/when it comes up I be very happy I've got it.