I'd argue that Netflix has these things that come out that grab people's attention all the time;crime docuseries, and those odd shows no one would really think they would like but they do, etc. People love those things. It's not Tiger King itself, it's just they can release these shows, usually only a handful of episodes, and it pulls people in. I can name multiple shows this year, last year, and years prior that pull people in just like Tiger King (Pandemic for example released this year early on in the coronavirus..kinda odd timing but perfect as well when you think about it).
Disney+ has some content, especially nature shows and a few odds and ends type shows, that can engage more than just those looking for Disney, Pixar, Marvel or Star Wars content but they won't necessarily have that pull that Netflix and premium channels can have because their content and intended audience won't allow for that. It's a downside to any streaming service that purposefully excludes certain aspects from the content they allow for.
I am actually concerned like you mentioned about future content. Not on Netflix and probably not premium channels because I think they'll be ok for a while (so really it's just that I feel they'll be in trouble later than others) but rather normal cable tv and streaming services that don't have content filmed from prior to coronavirus or content that they can purchase the rights to stream that is suitable for their intended audience. Multiple of the shows I watch have stopped early because they were still filming and producing when this hit. I am very interested in how the All Rise 'at home' filmed episode works out as that's supposedly a test for how it realistically works to film using people's homes and own devices and whatnot.