As a native American speaker, when I see British folk sign off "Cheers," I have always interpreted it as they are sending out a general well wishing. It does remind me of when Americans say "Cheers" when they click glasses in general goodwill to each other.
But more than that, whenever I hear a European say or write Cheers, I always think it sounds so cool and makes them seem so laid back, friendly, and even hip. I really love the phrase and always feel impressed when I hear it. I'm not sure why.
In fact, I wish I could say it without feeling funny. But it feels like cultural appropriation to say it. But I totally support other Americans saying it if they want to. It's just not a trend in our culture yet so it sounds a little funny for us to be saying it.