"Looking after the immune system" is just a nonsense statement.
If you're not vitamin or mineral deficient (and most people are not) the immune system has taken care of itself. You can't boost it.
What you CAN do is improve physical/metabolic fitness which is the UKs main problem - everyone is overweight and inactive.
If I was going to build a case for high vaccination rates just now I might look at US state-level data. States with high vaccination rates seem to be having a milder experience of delta both in terms of positive tests and deaths-with-COVID. Likewise the low-vax countries in Eastern Europe seem to be having a bad time just now relative to the rest of Europe. Which is not to say that a vax campaign focussed on risk groups, or heaven forbid, some early treatment, would not have been equally effective. Asia and sub-Saharan Africa might have been better off without vaccines.