Move complement setup stuff into the Synapse repo (#12404)

Fixes matrix-org/complement#330 (or it will, once we remove the old files).

It's not quite a lift-and-shift: I've also taken the opportunity to get rid of the custom CA that we used to use to sign the TLS certs, which has been superceded by the CA exposed by Complement.

.github/workflows/tests.yml

@@ -363,27 +363,11 @@ jobs:
             (wget -O - "https://github.com/matrix-org/complement/archive/$BRANCH_NAME.tar.gz" | tar -xz --strip-components=1 -C complement) && break
           done
 
-      # Build initial Synapse image
-      - run: docker build -t matrixdotorg/synapse:latest -f docker/Dockerfile .
-        working-directory: synapse
-        env:
-          DOCKER_BUILDKIT: 1
-
-      # Build a ready-to-run Synapse image based on the initial image above.
-      # This new image includes a config file, keys for signing and TLS, and
-      # other settings to make it suitable for testing under Complement.
-      - run: docker build -t complement-synapse -f Synapse.Dockerfile .
-        working-directory: complement/dockerfiles
-
-      # Run Complement
       - run: |
           set -o pipefail
-          go test -v -json -tags synapse_blacklist,msc2716,msc3030 ./tests/... 2>&1 | gotestfmt
+          COMPLEMENT_DIR=`pwd`/complement synapse/scripts-dev/complement.sh -json 2>&1 | gotestfmt
         shell: bash
         name: Run Complement Tests
-        env:
-          COMPLEMENT_BASE_IMAGE: complement-synapse:latest
-        working-directory: complement
 
   # a job which marks all the other jobs as complete, thus allowing PRs to be merged.
   tests-done: