The Kanamits don't seem to care about suspicion in the long term.
This is unfortunately a mostly speculative answer, as the episode itself has little to suggest their overall way of thinking beyond humans go well with a fine chianti and the right spices. There are a few pieces of information we can extrapolate though.
First off, and possibly the biggest, they already have a cookbook. This isn't speculative dining on their part, if they already have the recipes written up. It strongly suggests the Kanamits have already been harvesting humans for a fairly long time. Possibly, as you suggest, from populations where they won't be missed.
Second, they're pretty careless. The cookbook gets left. As you say, tourists will want to reconnect with their families and vice versa. You say don't say "no signal", but that's a plausible lie, and one they could use while accepting messages that they will "pass on" to keep people calm. Once the cat's out of the bag, the Kanamit with Michael Chambers even seems amused at his hunger strike. There's no way, as portrayed, they won't eventually get found out.
So you may want to assume that's the point. After all, their big push in the entire episode was to get us to accept technology that would improve our way of life. All things that would solve hunger, pollution... peaceful tech. If it was just thrown at us, humans would be suspicious and they wouldn't accept it. If it was done slower, with trust, we'd adopt it, and probably wouldn't get rid of it even if we later suspected the source, once it was useful.
So put all the guessing together, and what do you get?
Humans can't stop the Kanamits from taking people, because they've done so and will continue to do so for a long time. From the perspective of the aliens, this was just an operation to improve the quality of their herd animals, for finer selections in later harvests. And since they're already there... they did some early shopping.