I think this is one of the cases where GDP misses what we're trying to measure. If, before the Web, I would have bought a product just barely useful enough to pay for, then post-Web, buy something different instead that is much, much better for me (has a higher consumer surplus), GDP will be the same, even though human welfare has significantly increased.
Normally, you can glaze over such a case, but the internet is basically filled with this kind of thing. Moreover, a lot of the stuff that you would normally have to pay for, you get effectively for free, and I'm not talking about pirated movies. I mean the collaboration, fact-checking, knowing of more options, info you'd normally need to buy a book for, etc. Again, cash payments for that stuff might have dropped, but it made people that much better off.
So if people that were doing things obviated by the Web, then go start doing something different, then, well "GDP is GDP" -- but that is a massive efficiency improvement.
I don't know about human lifespans or manufacturing output (though the widening collaboration opportunity increases the number of people that can solve foundational manufacturing problems), but that's only one of many human values, and I think it's an artificial requirement to compare to electricity on that basis -- almost like penalizing it because it didn't enable better hunting of large animals.
So stepping back to the level of human welfare and modes of life it enables, I think the Web is comparable to electricity.
Here's an informal measure: what does the technology do to the "quaintness" of (non-sci-fi) stories composed before it? After electricity, you might look at a book with a plot element "character can't work at night", and scoff at how backward it is.
Does the Web do something similar? I'd say it does a lot more. Per Steven Landsburg, there was a novel written in '91 (right at the Web's infancy) with plot elements like "someone is endlessly searching bookstores to find an obscure book" and "someone sells expensive encyclopedias to people with a tremendous demand for easy access to knowledge" -- very, very quaint from today's perspective.
(Novel was called, IIRC, "Hunters and Gatherers")