Yes, but testing takes time. Automated testing is a wonderful thing, but it only catches those things that the testers have already though of AND have had the time to write automated tests for. Some things are notoriously difficult to automate testings of. And there are always things that don't get considered during the writing of tests; especially those things that only appear when multiple systems are interacting together.
Taking the time to deploy code to a staging server, where QA can check it, and other systems can interact with it, can prove to be very beneficial. And runs contrary to having every push (or every push that passes the automated testing suite) automatically go to production.
I'm a big fan of something like a 2 week release cycle, plus the ability to release quickly when needed (generally same day, but only when necessary). I'm a big fan of automated tests (both unit and integration, both dev and QA written), plus a QA person/team to bang on things in a way developers don't think of, plus a staging environment where all the (staging versions of the) system interact with each other.
I'm not a fan of pushing things to production willy nilly, with a high risk of breaking things "because you can push again with a fix in an hour".
