If I republish, yes.

If I just download a site – for local use, or similar cases – it’s not an issue.

And if I provide a scraper where the user customly has to enter the URL, it’s also technically just a very advanced webproxy, and also okay.

That is incorrect. Even if you don't republish the data, the scraping alone can be a violation of the site's Terms of Service, and you can be sued (and lose). See the Wikipedia article I linked above.

Think about it from the point of view of the site being scraped. Suddenly your webserver is being slammed with traffic that's way beyond what any human would cause. It's costing you bandwidth and compute resources. You have every right to say no, scraping is not allowed (in your Terms of Service). It's then up to a court to decide if those Terms of Service are legally binding. Some courts have decided that yes, they are.

And where I am, courts decided that no, ToS are never binding. ToS are effectively worthless.