It's probably part of how they keep things simple. But you seem to imply that storing lots of data is hard. It doesn't have to be. I've worked with search teams that operate at a very large scale. You need lots of hardware but not necessarily lots of people.

The scale of Twitter is large but not enormous. Basically a few trillions of messages overall. Most of them pretty small. All of them immutable (well so far). And lots of images/videos with some CDN in front of that. It's a lot of bytes but not a whole lot of complexity. And a couple hundred million users and user profiles. I can think of a quite a few ways of dealing with all that.

What's hard is things like algorithms and good search. But that's a space where you can achieve a lot with a small and focused team with access to lots of hardware. And obviously this is also something where Twitter has maybe been a bit underwhelming and struggling. User growth and engagement stagnated years ago for them. This was not a healthy social network. It's not like they were nailing this. Throwing more people at the problem wasn't working.

I believe back in the day the messages were stored on the users phone, but now with the web version, i think they might be storing the data now.