Well, they did, and the whole wasting-civilians-from-the-past-to-fight-a-lost-battle-thing was there to win time. They managed to find and to send the poison back in time. It was the story of the movie that those beasts appeared to be unbeatable - until research found out, how they could have been beaten if only one would have known earlier. Now send that information back in time - voila, humankind prepared. They choosed not military but people that wouldn't have any more children in the future which was the past of those coming from it. That's to "avoid paradoxes", quite reasonable. As those guys and girls are more likely not being military personal due to age, that's why they choose civilians. Some with a military background. That's it.
"Abysmal plot holes"? I don't think so. One may doubt that humankind will find a way to create a wormhole to travel back in time within a few decades, half of the time fighting a lost war on the way. That was never explained in the movie. My major question during the movie was how could those beasts that obviously hasn't anything else in mind than finding food be a space-travelling species? That was explained in the movie: they were the cargo of a space-travelling species.
"What was the point of the toxin if they could just kill them with explosives?" That's when they know where to put the explosives. The people in the future had never discovered the crash site of the alien spaceship, they did not even know that there was one. But the information that enabled the guys in the past to do that was as well obtained from the future, by chance. The guys from the future came up with the toxin plan. That finding a crash site of a single spaceship within the arctic and blow that up could be a solution as well they did not know and could not assume. They had absolutely no clue how the beasts came to earth and that was mentioned several times in the movie. Once the beasts were discovered there were too many of them already due to their high reproduction rate. That war was lost at that point in time already. One may ask wether a big and complex beast can reproduce so fast - questionable from a scientific point of view - but now we are discussing details. This and the wormhole were the "Let's assume it may be" points in the movie.
H.G. Wells once explained how to write a phantastic story: choose a phantastic idea to be real - like pigs could fly (he really used that as an example) - stick to it and follow the consequences.
I think the authors of "The Tomorrow War" just did that.
