fix: use node:20-alpine in sync builder, matching backend Dockerfile

node:20-slim picks up the Apple npm proxy which blocks installs.
node:20-alpine does not. Also add PATH for node_modules/.bin so tsc
is found, and remove committed dist/ since the build now works properly.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

.gitignore

@@ -10,11 +10,10 @@ yarn-debug.log*
 yarn-error.log*
 pnpm-debug.log*
 
-# Build outputs (sync/dist is committed since server can't compile TypeScript)
+# Build outputs
 dist/
 build/
 out/
-!sync/dist/
 
 # Database
 *.sqlite