Apparently they hunt and kill deer and reindeer in Finland.[1] I would have thought the same but seeing your comment prompted me to do a search. Found a handful of YouTube vids claiming the featured Lynx killed the deer they were recording, but none of the first 5 I checked featured a Lynx performing the kill, only the eating.
Google says there are 100,000 deer, 200,000 reindeer and ~1200 lynx. So there's enough that they could make a difference (but it matters a lot how often they would take a deer).
Turns out we suck at engineering ecology: about the best we can do is introduce a wild population into conditions we think they’ll survive and let nature take its course. Prey and predators eventually reach their own sort of equilibrium if we leave them to it, absent factors which are severely detrimental to either.
And this isn’t a deer extermination project either: it’s a lynx reintroduction project.
Ed: I stand corrected, I guess.