There's an obvious difference between a browser that, in normal use, doesn't allow storing of streaming video, and a tool that does and isn't supported by Google at all.

Yes, you can download the full video with a browser. Yes, you can use wget or curl to do that. But youtube-dl's community was dumb enough to give clear cut examples of downloading copywritten material, which is obviously not what the RIAA or probably even YouTube's lawyers want.

Courts care about intent, and it's a straw man to argue that technical similarities mean they're half-assing their enforcement.

You can right-click->save-as the video link in the network panel of dev tools of either Chrome or Firefox. Google's own browser lets you save youtube videos with a few clicks. Also taping TV and radio is legal as well, even though the receiving device doesn't offer that functionality, unlike Chrome.

Is there a law that says https protocol has to be consumed by a browser?

Someone should issue DMCAs for the Chromium and Firefox repos. Fight fire with fire.

How is it illegal to file a DMCA takedown request if you believe you have legal standing to do so?

EDIT: The DMCA doesn't require that you have a law degree to submit a takedown request, it just requires that you truly believe it is infringing.

You also need to either own the copyright or have permission from the owner.

My understanding is that it only requires you send the notice in good faith, meaning that if you think you own a copyright but don't (because, say, you own too many copyrights to keep track of them) you're still in the clear.

Apparently not, since people get away with sending fraudulent takedown requests all of the time. Now that I think of it, has anyone ever been punished for abusing the DMCA?