If you get insurance through work, it pays to fix you regardless of where you get hurt. There may be co-pays and deductibles, but then the insurance pays. Go on ya for not having an LTA (lost-time accident). Companies need to keep these stats low, so you're doing them a big favor for not leaving early (as if they'll ever show their appreciation -- the bastards!)

There’s a difference between health insurance through work, and a company’s liability insurance.

This. My mom worked in an office answering phones and would file a report for anything as small as a paper cut 😂😂 She said if anything happened like an infection or something, she had it on record. This is probably on the extreme end of things lol but it definitely instilled in me the importance of documenting things for liability, etc. In your case...yeahhh you slice your leg open at work, you definitely file a report and have them cover it! Maybe you took pictures, have witnesses, maybe there’s cameras...something you can do to still file a report and get reimbursed. Good luck, and kudos to your strong work ethic! Balance it with self-care, and you’re good to go :)