The headline of that CDC article is misleading: the actual study compares the protection given by being previously infected with that given by being previously infected and then also vaccinated. There's no comparison against people who're relying only on vaccination to protect them. This is also a pretty weak justification, since if I remember rightly there's a comparable improvement by using one of the more effective vaccines rather than one of the less effective ones or possibly by increasing the number of shots required for a complete course of vaccination - but there's a huge taboo about noticing the difference in effectiveness between vaccines. (Also, I think this mandate kind of incentivises people to get the least effective of the US-approved vaccines, since it's a single shot and that makes it quicker and easier to meet the requirement.)