I think a lot of you are grossly misusing the concept of a plot hole. This is a fairly common mistake.
A plot hole is not “I can think up a better way they could’ve done this” or “their plan was sub-optimal”
Using runners to deliver messages in WW1 was extremely common. Using planes to that effect was uncommon. It is not only completely plausible that a mission like this would’ve been assigned, but a novel solution like air dropping it would’ve been the unusual scenario that required special set up.
Additionally, this movie is very much explicitly told from the perspective of a couple of low rank enlisted men. We are only shown their perspective. For all we know, in the general’s tent before we got there there could have been a discussion about how to best deliver the message. Maybe they did consider air dropping it, or trying to find a unit in the area with a radio, or some other method. Maybe there were perfectly good reasons to rule them out. Since we only see what the grunts see, there’s no way to know.
Additionally, maybe they actually are trying to reach the attacking force via other methods, but feel as though using multiple methods would yield better results. They wouldn’t necessarily tell that to the runners, to impress upon them the importance in their part in all this and motivate them.
You guys are vastly underestimating the complexity of running an army of millions of men. Organizing them and keeping them all on the same page is enormously complex. Communications are broken all the time. Even today, you can have two military units right next to each other who can’t talk to each other because they’re not part of the same chain of command and have incompatible radios or some similar reasons, let alone a hundred years ago. Armies at war are enormously complex beasts and WW1 was extremely early for a lot of modern communication methods and techniques.
The process of getting recon from planes and then disturbing that recon, acting on it, and then communicating with random moving unit on the attack is way more complex than you think. There are chains of command and communication that dictate the use of assets like airplanes. The general we saw may have first have had to contact a general up his chain of command who could then contact someone who liaisons with those planes who could then send the message to those planes who could then use rudimentary air navigation skills to try to find a unit they can’t communicate with to drop orders that might land in no mans land or get lost in the mud. Or shot down or even just driven off along the way. In no way is that a trivial mission to arrange and pull off. And depending on how many command chains you had to through, and what the status of the communications at the rear were, it may not even be faster.
Sending runners may have been the best plan available, or only plan available. Or maybe he did try both, but our grunts think there the only method as to increase their motivation to complete the mission. So no, this isn’t even remotely close to being a plot hole, and it may have even been the best plan given the constraints.