They decided to keep the code on the local machine, but the language server on the remote one. That seems like a recipe for inconsistency. You only get relevant results from your language server once your code has synced.
The article mentions that the LSP itself already has baked-in support to enable editors to send chunks of unsaved edits to the language server (LS) as they happen.
What Stripe’s configuration introduced is that they used a remote LS instead of the default local LS. Regardless, VS Code already defers LSP communication until it feels idle, and developers are used to that. So I wouldn’t expect a remote LS to significantly impact the level of inconsistency that developers already accept when using a local LS.
On the other hand, there was so much code that running everything on your own laptop was essentially out of the question. Doing a git pull after a long vacation locked up your dev box for a hot minute while it checked all the types—doing the same thing on your MacBook would be painful at best.
The code syncs on every keystroke. Consistency isn't an issue unless you are having connection issues. And if you are then pretty much all development is broken anyways.