I can almost guarantee you that a synchronous API would yield a much more complicated design just because many operations can not be expected to reliably work within a few milliseconds.
You want to rename something across multiple files? Well now you add file system overhead and blow straight past reliable frame timings. Good luck waiting for that operation to finish. Of course you could make the API beginRename, and queryRename, or whatever you fancy to see if the operation was successful, but now you're back to what you wanted to avoid: an asynchronous API. Do mind that the example is actually one of the better cases, as many things you might want to do with a codebase are actually more expensive. You will feel the hiccups from waiting in the UI thread and you will loathe the program for it.
Would you write your editor that way if it didn't use LSP but was specific to one language? If not, but you would write it that way with a language plugin, why? You still have to write a good product. You can't blame the plugin API for everything you get wrong.