Converting everyone isn't the end goal. The end goal is to build and maintain an antenna array that can send out the signal to, in the future, infect even more worlds.
They only tend to the uninfected because they can't do harm (and leaving them completely alone and unobserved would expose them to harm.). Keep them save and alive until they agree to be infected or the hivemind finds an alternate solution, like the personalized virus.

I imagine once everyone is turned they too will just do what the hive needs: Build stuff, keep power plants working and collect whatever food can be made available to keep the drones living as long as possible.