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.
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).