One thing pointing out is that deer aren’t really the disease reservoir for Lyme disease (it’s primarily mice) - but deer are key for tick population growth.

Mice and voles, yes.

(There are more voles in the UK than mice, as it goes. Outnumber them by about thirty million or something. Fairly recently acquired trivia for me, but it turns out that if you see a mouse scurrying in the wild in the UK, away from houses, it's a bit more likely to actually be a vole.)