Upgrade to TypeScript 7

Author: cmoesel

DEPENDENCY_NOTES.md

@@ -3,5 +3,4 @@ The following notes about the app's dependencies should be revisited and resolve
 As of 2026-07-27:
 
 - `@types/node`: Consider updating to `26.x` in the future, but for now `24.x` still has plent of time in LTS.
-- `typescript`: `7.x` uses a new native compiler without a native API. Consider updating soon after evaluating the impact.
 - `tslog` (test-server only): `5.x` is ESM-only and replaces v4's flat logger settings such as `hideLogPositionForProduction` and `prettyLogTemplate`. Updating will require some migration.