The suite is a gate
A red suite should block the merge or the deploy, not be a number you glance at and dismiss. Wire the exit code into the build so a failure cannot be ignored. The discipline is that shipping with a red suite is not an option, which is what keeps the wall honest over time.
Run it where it counts
Running locally proves nothing about the server. The same suite must run in CI, on the real environment, on every change. That is the habit that turns a suite from a local aid into a permanent guard, and it is where the works-on-my-machine bugs get caught.