I think what's interesting about Lean is that Lean is a language, and what most people are talking about is a library called Mathlib. From what I've read about Mathlib, the creators are very pragmatic, which is why they encode classical logic in Lean types, with only a bit of intuitionistic logic[1].
[1] for those unfamiliar with math lingo, classical logic has a lot of powerful features. One of those is the law of the excluded middle, which says something can't be true and false at the same time. That means not not true is true, which you can't say in intuitionistic logic. Another feature is proof by contradiction, where you can prove something by showing that the alternative is unsound. There's quite a few results that depend on these techniques, so trying to do everything in intuitionistic logic has run into a lot of roadblocks.
[1] for those unfamiliar with math lingo, classical logic has a lot of powerful features. One of those is the law of the excluded middle, which says something can't be true and false at the same time. That means not not true is true, which you can't say in intuitionistic logic. Another feature is proof by contradiction, where you can prove something by showing that the alternative is unsound. There's quite a few results that depend on these techniques, so trying to do everything in intuitionistic logic has run into a lot of roadblocks.