We are serving the content ourselves, but we fully attribute it to the source and provide a link to the original website. We aren't doing anything much worse than for example the browser cache does when loading up a cached website.
The difference between what search engines/the Wayback Machine do and what a service like his does is that the former caches more of less complete copies of the content without the intent to strip anything out (save for the view as text only feature in Google's cache), while his service is stripping out the monetization schemes of the content owners.
Although he links back to them (which he absolutely must do anyway), the content owners might become angry that they aren't getting the traffic and the chance to have their ads clicked. I think that is a reasonable viewpoint for the content owners to have, yet I also want to believe that a service like his should be able to exist. Perhaps some kind of middle ground can be achieved where he does show the ads but perhaps well organized at the bottom or middle of the article or something.
What would the content owners be able to do though? Maybe send DMCA notices. This would fall under copyright infringement laws I think. I'm no lawyer though. I also think there should be some kind of allowances for these kinds of content repurposes.
What about copyright legislation? As a rule of thumb you should assume that you are forbidden to do anything with other peoples content that isn't explicitly allowed.
I know law changes/not everyone sees it the same/different courts rule differently, but if one accepts that taking an article in full and republishing on another site (even if sourced/linked back) without permission is a copyright violation, I'm not sure that you doing the same thing (and on a wider scale) isn't also a copyright violation, even if the final article is presented on a different medium.
I'm so very much not a lawyer (pretty obviously), just someone with a publishing background, but it might be worth you talking with a copyright lawyer just to make sure you're on solid ground before you launch it with the content/way you're currently planning.