Ok, so I'm a lot late to the topic, but I found out that if you change the date to, let's say, before December 31 2020, the website still runs. It didn't work on Chrome for me, but I tried Internet Explorer and it ran just fine.
Alright I'm waaaaay too late but after removing http://nisamerica.com/games/cave-story/ from unit_top.swf>scripts>DefineSprite (83)>Frame1>DoAction and putting the music files inside a folder named sound it restored the BGM button.
Using inspect network activity I also found out that the Story, Reviews, Characters, System and Media tabs have their own swf files.
Now I guess the website is preserved, click here to download it.