Once I installed the base LSP package, and restarted Sublime, then it gave me an error[1] at startup, explaining that it couldn't initialize the LSP for TypeScript. The error message included a button to fix it.
After that, yeah, it works a lot better! Jump to definition works, and the as-you-type errors I mentioned before now do appear. Also, the auto-complete list looks fairly sane, without all the "random guesses from all the text in this project" entries it had before.
So... not bad! At a glance I think it is still probably not as good a VS Code, but it might be usable. I will give it a closer look later.
[1] The error was: "Could not start LSP-typescript due to not being able to find Node.js runtime on the PATH. Press the "Install Node.js" button to install Node.js automatically (note that it will be installed locally for LSP and will not affect your system otherwise)."