Janeway’s primary mission is to get the crew home. If I remember correctly (which I may well not), at that point, she believes (correctly) that Starfleet considers Voyager destroyed in the Badlands, and thus isn’t searching for them, or working on ways to bring them home.

If she can inform Starfleet that they’re actually alive and stranded in the Delta Quadrant, then she no longer just has her ship and crew working to get Voyager home, but potentially the best minds in the whole of Starfleet too.

Sure, losing the Doctor would increase the risk of some or all of Voyager’s crew dying due to a medical skills deficit on board. But there are plenty of other unrelated death risks scattered around their long route home, which they might avoid once Starfleet is on the case — for example, the Borg (as you may remember from *Scorpion*), who aren’t going to be particularly slowed in assimilating the entire crew by an emergency medical hologram (see *Star Trek: First Contact*).

And, in general, the risk of dying for the sake of the mission is something that all Starfleet officers sign up to when they join.

It’s certainly a gamble, and a bold one. But it’s exactly the sort of gamble that a captain needs to be able to take. This is why Picard never joined the Enterprise senior staff’s weekly poker game; he, like Janeway, would crush them without blinking.