build: fix build environment for the liveness test (#10551)

Extracted from #10210.

Make the test more reproducible, so that it does not require coordination
between the build container and the run container.

- Use layers to more advantage.
- Include bash in the run container.
- Put writable output onto a volume.

.github/workflows/test.yml

@@ -228,7 +228,7 @@ jobs:
 
   liveness-test:
     runs-on: ubuntu-latest
-    timeout-minutes: 10
+    timeout-minutes: 15
     steps:
       - uses: actions/checkout@v2
       - uses: actions/setup-go@v2.1.4