In my view the main thing that's not fun about a cheap bike is if it malfunctions. But there are two other things as well. As others have mentioned, fit is important. If the bike is really uncomfortable to ride, then it will never make it out of the shed. The other thing is the tires. There seems to be a growing consensus that somewhat wider tires are preferable for commuting, so at the very least I'd avoid a road bike that only accommodates really skinny tires.

But plenty of people commute on old road bikes.

My view is that there's no way to guess exactly what you will want, and the only way to find out is to start with something, get riding, and think about what your next upgrade or next bike will be.