My mother was the same way. Finally not only got her to try Medium cooked steak, but now see her ordering it more than once on her own when we go out to catch up. She still thinks I am crazy to eat mid-rare to rare but I like it that way. To each their own I guess....
I can’t eat well done meat. Fooey. I love a good rare steak, but started eating it medium rare when they were warning about getting sick from meat here. If I can get good meat, I’ll broil it rare-ish. I can get some pretty good meat from farms round here, but it’s pricey.
I’ve had steak tartare, it was yummy.