A lot of stuff that's "fairly well agreed" is wrong, or at least misleading. This is one of them. Text can be made attractive and inviting, or not, information can be easy to find, or not, according to how well the page is designed (or not!).
People will read if they are interested and you don't actively drive them away with dense, hard to follow text. A page full of bullet points can repel the reader just as thoroughly as a solid screen of undifferentiated copy.
There are three things you need to achieve with a web page, a flyer or an ad:
You must attract the viewer's attention (some marketing people say "interrupt" -- same thing). Have you ever tried to talk to someone without first getting their attention? Wasn't successful, was it?
What they see as an image or read as a headline must interest or engage them.
Then you can educate, deliver your message, or whatever you need to get across in your text.
You have less than a second to achieve 1, and only a second or so thereafter to achieve 2, but once those are accomplished you can expect the viewer will read at least the first sentence or two of your text. Those sentences, well written, will draw the reader into the rest.
A very large part of a designer's job (applies equally to book design, web page design, any sort of message) is to make the information attractive enough, and findable enough, that the reader feels comfortable and invited. How much text and how it's written depends very much on the audience, and is the job of the copywriter.