I think the problem is that you imagine "infection" to mean "disease" or something similar. Infection means that the pathogen gets in your body and starts to grow. There is no vaccine that can prevent that. What a vaccine can do is to cause your body to respond to an infection more quickly and more effectively. If you're lucky, the response comes so quickly that you never knew you were infected.

So that aside, what did you mean exactly? When you said

> Much protection against infection or against hospitalization?

Did you mean to say: "Much protection against asymptomatic spread or against hospitalization?" Or maybe you meant to say something else?

No, I meant symptomatic spread. I don't care much about getting sick. I worry about winding up in the hospital. As far as I know, the 3rd dose of Moderna and Pfizer are both effective at keeping me out of the hospital. I don't know the exact rates, but I've met plenty of people who had mild illness recently.