Simplify language server startup (#6449)

These are just some small refactorings of our language-server-starting
code, motivated by another change that I decided to bail on:
https://github.com/zed-industries/zed/pull/6448.
6ed7cc7833Max Brunsfeld committed on 1/25/2024, 1:36:50 AM· committed by GitHubparent90c1d8f
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