I suggest doing a test run of the actual commute during the weekend, which should resolve the mental resistance.

At first, start out early, like 30 minutes or an hour, to give yourself time in case something happens.

Mechanical issues other than a flat now and then can be mostly avoided and are usually pretty minor. If you pay close attention to the bike and maintenance, which I've found to come pretty naturally when you're commuting frequently. I'm more inclined to check everything over and deal with whatever comes up before it becomes an issue. Nevertheless, when I started I felt the same way and would take a lot of tools with me, even random replacement parts if I had them. There's nothing wrong with that, though eventually you pare it down to only what you really need.