There is a downside. The faster, and more often, you release changes to your production system, the more likely you are to cause an issue
The evaluation needed here is that the potential reward of a more complete correct system out weighs the risk of causing an issue. If the perceived gain is minimal or only cosmetic, there might be more risk then reward. But coupling that update with another of higher value, the pair would be worth the risk. Again the risk should be mitigated by proper testing, review etc.
The CD approach has within it a goal of validating and testing the new software/process to a point where the risk in minimal, this being done quickly and accurately.
If a company is willing to make the investment to mitigate the risk accordingly, then CD is a great approach. Without it CD is only getting errant software out there faster, which no one wants.
Not all gray hairs are Dinosaurs!