Like yourself, I have given up on the generic newspaper market which aims to be all things to all people.
However, I have replaced reading the generics with reading the specialists.
I get more - and generally better quality - writing from the likes of The Economist, New Scientist, etc. I occasionally dip into the more political magazines to get some broader opinion based writing, but it is a choice to do so.
To have personal opinion slotted in between serious news is not just distracting, but actually quite annoying as I like my news to be factual, not opinionated.
I still like the print format for serendipity discovery though, something which is harder with reading publications online as it is often easier to skip headlines that don't appeal. As you also noticed, in print, you at least read the leading paragraph of each article.
I look forward to more magazines coming out in e-book format though, as I can resize the fonts at will and not have to rely on my glasses to be able to read the magazine.
Maybe then I can have the best of both worlds - the page turning disciplines of the print format, but with some of the digital tools that make reading online so much easier.