Problem

TypeScript 5 beta is now available: https://devblogs.microsoft.com/typescript/announcing-typescript-5-0-beta/

We should directly update to it for JupyterLab 4.0 since its final release will likely be available before JupyterLab 4 final.

Proposed Solution

Update to TypeScript 5.

Additional context

Some work to update TypeScript had already started in #13028. Although it's been lagging a bit behind and had to go through multiple rebases due to issues in some dependencies.