If it was me, yes. But keep in mind I am quite temperamental, and (as much as I hate to admit it) I get pretty "emotional" and "offended" for about two weeks every month

. And am mentally pretty unstable, so yeah.
As for a hair appointment being a "masculine" or "feminine" thing, I've liked them since I was able to choose what I wanted done with my hair. It gives me a "fresh" feeling. When my mom still dictated what my hair was like as a kid, I hated it. On the other hand, my sister (quite girly, but not the type to wear dresses everywhere) never likes getting her hair cut or hair done up any special way and prefers it just to hang naturally. With my dad it's a 50/50 if he is happy or annoyed, but that has more to do with the hair dresser. If the hair person doesn't suck, he is generally happier afterwards. And my mom is always happy after a hair appointment while my grandma thinks of it more as a nuisance. I think it varies per person, and would probably have been more angry by how sexist that statement was.
And at least from pics I've seen, testosterone won't make you look like a "butch woman" in the end. In the beginning maybe, but most likely not after a few years of use. I think your relative most likely isn't aware of what testosterone can do to a person. Most people aren't going on the internet and looking up "FtM Testosterone Before and After" pictures for example, even if someone close to them is going through this stuff.
Keep in mind I'm pre-everything right now and am not sure if I want to take hormones, so this is just my viewpoint if either of these situations happened to me.