[ci] Use bdt CI templates more, even for test-only phases

6 jobs for update-ci in 65 minutes and 50 seconds (queued for 2 seconds)
Status Job ID Name Coverage
  Build
passed #157608
docker
build_linux_36

00:20:11

passed #157609
macosx
build_macosx_36

00:47:34

 
  Browser Tests
passed #157641
docker
browser_test

00:18:16

failed #157629
docker
browser_test

00:08:59

failed #157623
docker
browser_test

00:03:50

failed #157610
docker
browser_test

00:03:31