Yes, but like I said, I'm not sure how you test for those so I don't know how you prove who has them, unlike our current testing capability for antibodies. I'm not arguing, or even saying it's not possible. Way back in the early days of the pandemic, there was talk of T-cell stimulating vaccines.
And the "infection" vs "Covid" discussion is circulating again. Having "Covid" implies symptoms. Having an "infection" means the virus is in your body - it doesn't automatically mean you have symptoms, but it does mean you may be capable of shedding live virus to infect someone else. The vaccines prevent most symptoms, i.e. Covid, because that's all the studies were designed to show. They are still studying if they also prevent infections as well. I seriously think it will bear out the vaccines do prevent infections and therefore transmission, once more time and good research has been done.