I was there often on Shabbos and it was very weird davening at a tourist site- but if you ignored it, it was awesome.
I was also there Yom Kippur (singing leshana haba… not at the kotel but as close as you can be to it basically without being there, which was a bit surreal), and while I was walking to davening before neilah, I saw a crowd of tourists sitting on the plaza with their cameras out, waiting to watch the Jews do their ritual. It was… interesting. I was glad I was davening indoors.