That's small "r" republican, it's just in the title so tradition dictates that I capitalize it (yes, that is meant to be ironic).
The British Royal family is just background noise for me. It gets mentioned in American news often enough that I just think of it as one more weird thing the British do.
But they're pretty much the only royal family that Americans get regularly exposed to, so I think a lot of us (well, at least me) go through life with the impression that the civilized world has done away with that particular barbarous relic and the British are just weird.
Then every once in a while something like a marriage in Luxembourg reminds me that this is not the case. I know in my head that lots of modern societies still have royal families and I know that none of them have any real power, but I sort of forget about it (except in the case of the British) until news items like this pop up.
And then it bugs me.
The French were admittedly a little excessive. There was no reason they couldn't keep their heads (although it wasn't the worst crime in history). But every self-respecting modern society should have done away with all this by 1918 at the very latest. There's certainly no excuse to let a fellow human being prance around with pretenses of nobility and royalty in the 21st century.
Or put it this way - in the 21st century, what do civilized societies do when people run around proclaiming themselves to be royalty? We give them a dollar so they can buy some food to take back to their cardboard box, or if they are a particularly bad case we put them in a mental health treatment center.