Was I incorrect that he was talking about small sites, or do people just believe that every site regardless of size should use a deploy script and the other trappings of large application/high volume production development? Because the latter is silly.
Doing a tested change and then a pushing to production is a 4 command overhead using git as lazily as possible with maybe an additional 5 minutes spent during initial project setup. You're doing the same coding work, you're just helping to ensure it's not going to create any completely useless work spent undoing mistakes. It's an ounce of prevention, pound of cure argument in my mind.
It's really not that clear cut. To do what you propose, you either have to set up an identical staging environment, or you have to set up a local environment that mimics (and stays in sync with!) the production environment. For a tiny site that sees <100 people a day, where the potential damage of a mistake is absurdly low (so what if 5 people see the broken version while you make corrections?) it's not necessarily worth the additional hassle.
This all depends on what kind of site it is, of course, but in the context of what he's saying, it makes sense.
