This is probably illegal in the United States. Expect to receive cease and desist demands.
It's also breach of contract (which I'm labeling separately from "illegal" to avoid nitpickers, even though it could be included) due to the individual ToS on each site, which almost always include boilerplate forbidding the access of the site by "automated means" in addition to forbidding "commercial" or other non-personal use.
Before you raise the common counterarguments, please know that others have done so before you, and the courts have generally sharply disagreed with them. There is no respect for non-Google data scraping in the judiciary.
Usually if you C&D upon receipt of C&Ds, you won't get sued unless you actually damaged the site. I assume companies like import.io and Scrapinghub adhere to those. I know particularly in the case of Scrapinghub, they won't go behind a login to get something in order to try to avoid heat.
Fundamentally, however, anything that makes scraping a key component of its business model is a high-risk business under current US law. People can and have sued scrapers out of existence, often leaving a trail of screwed customers, laid off employees, and dejected founders owing huge liability judgments.
It's weird that the comparison between posting a robots.txt and a "no trespassers" sign hasn't been upheld? Or has it.
IMO the tort of copyright hasn't kept up well with tech changes, but transient cache copies are handled in EU laws IIRC.
Similarly with your mention of the CFAA (UK CMA has some similar terms), they're very loosely drafted. Not accounting for the need to communicate the limits of allowed access is silly though; there's a presumption of allowed access online IMO (that isn't mirrored offline) and going against that presumption should require explicit withdrawal of consent.
If I were drafting the law ...