> That syntax tree still needs to be stored in file(s) somehow

Sure, but there are less flaky ways than spreading a syntax tree across files. Visual Age actually used a database for this back in the day. Smalltalk did similar things by storing code in an image file that contained both byte code and method definitions. You could export source code if you wanted. But wouldn't do that while developing typically. That's not an approach that caught on. But it has some advantages.

What you are describing is what Eclipse did with Java. Eclipse was the successor to Visual Age. The Eclipse incremental compiler for Java updated an internal data structure for the IDE. It could do neat things as partial compilation to enable running tests even in the presence of some compile errors. It also was really fast. By the time you stopped typing, it would have already compiled your code. Running the tests was similarly fast.

The problem of syncing a tree of source files with an AST is just a bit hard. Intellij never came close to this and has always had lots of trouble keeping its internal caches coherent. There's even a top level "invalidate caches" option in the File menu (still there, I checked. Right next to the Repair IDE option). They were off by 2-3 orders of magnitude. Seconds (at best) instead of milliseconds. I still miss Eclipse's speed every day I use Intellij.

Some compilers are taking some steps to supporting more advanced IDEs. But there aren't a lot of those beyond what Jetbrains provides. VS Studio Code support varies between different languages. But mostly it's very limited on this front. The Rust compiler is one of those. Though I don't know the current state of that. Mostly it's not well known for its blazing performance (the compiler). I'm not sure if Jetbrains leverages many of those features in its Rust IDE (I'm not a Rust developer).