Generally agreed on just about all points made here, although aside from Barrow and Tipler, I have rarely run across these bad interpretations of the anthropic principle inside scientific circles.
“People claim that the anthropic principle supports a multiverse”
I haven’t heard people claim this so I wonder if it might be a straw man. The claim I have heard (which I would agree with) is that if it’s difficult to get inflation to happen without it being eternal, then we should take the idea of a multiverse very seriously — possibly even considering it a generic prediction of inflation. And if we combine that with anthropic considerations, then we get a very nice satisfying explanation of the measured value of the cosmological constant.