Right now I think it's possible to introduce lint failures (or Python unit test failures) by committing directly without Travis checks (example).

Admins can push without passing Travis checks, though we rarely do. We could potentially just require admins to pass Travis checks first? It's sufficiently rarely used it's unlikely to make much difference. @jgraham?

AFAICT, we're not currently continuously running these tests for the master branch, but I've seen for other GitHub repos that they can have a current build status indicator (https://docs.travis-ci.com/user/status-images/).

We do already run tests on master (indeed, it's the default Travis setup). We just rarely act on it failing. See https://travis-ci.org/w3c/web-platform-tests/branches.

Ah, neat! Do you think the reason why people don't act on it failing (which is rare) is partly because it's not obvious?

Maybe we could put the little build status badge thing on https://github.com/w3c/web-platform-tests/blob/master/README.md or maybe https://pulls.web-platform-tests.org/?