In several European countries until 2003 or so, this was indeed the case.
For example: Making a local temporary copy in a cache or on a hard disc counted (obviously) as making a copy under the U.K.'s 1988 Copyright, Designs, and Patents Act, which had no exception for computer caches and temporary copies, and which pre-dated the invention of the WWW (albeit not of UUCP). Doing so thus infringed upon copyright.
* https://legislation.gov.uk/ukpga/1988/48/contents/enacted
The European Union recognized this defect in the law and came up with Directive 2001/29/EC, which explicitly carved out in article 5 an exception to copyright for "temporary acts of reproduction".
* https://eur-lex.europa.eu/legal-content/EN/TXT/HTML/?uri=CEL...
The U.K. implemented this Directive (Directives having to be implemented by legislation in member states of the E.U.) via a statutory instrument, the 2003 Copyright and Related Rights Regulations, which amended the 1988 Act to carve out this exception in U.K. law.
* https://legislation.gov.uk/uksi/2003/2498/note
* https://legislation.gov.uk/ukpga/1988/48/section/28A
Browsing the WWW with caching proxies and caching WWW browsers was technically a violation of copyright in U.K. law for about 11 years. And the U.K. wasn't alone in this. Several other E.U. member states also had to tweak their copyright legislation because it didn't carve out this exception for cached WWW browsing.
But the RIAA's objection to youtube-dl is not the making of temporary copies for legitimate browsing and viewing purposes. It is the circumvention of YouTube's mechanisms in order to make permanent copies that aren't being used as an integral part of WWW browsing. The RIAA is not challenging the making of temporary copies. It is challenging the circumvention of access control mechanisms.
It is asserting that youtube-dl circumvents the "YouTube rolling cipher" mechanism, whatever that is (the referenced explanation being in German and apparently gibberish), and so is a tool for circumventing technological access control measures. So whether you can legally use WWW browsers is not the point at issue. It is whether youtube-dl is a tool that enables access to YouTube contents by circumventing a "cipher".
* https://openjur.de/u/2194436.html
Ironically, this same consideration applies to tools like
* Real Player Downloader (https://www.real.com/uk/realplayer );
* Media Human's YouTubeDownloader (https://mediahuman.com/howto/download-vevo-video.html );
* Ace Thinker's Video Keeper Lite (https://acethinker.com/video-keeper-lite );
and many others. Interestingly, furthermore, the RIAA appears to assert that the "rolling cipher" applies to all YouTube content, however copyright licensed.