Yeah, I know that his main goal's doing it for the populace, but they're still giving the military special training for the crossbow (Oda Nobunaga's tactic) specifically because of its downsides, in which case I feel like he could have also just manufactured a version of a compound bow for the army, so they could operate at their normal potential, rather than having to adapt a strategy meant for guns for their crossbow usage.
The strategy is good for guns because guns were substantially more powerful than bows and so building a strategy to mitigate their disadvantage was still better than bows, but if a compound crossbow and a compound bow have similar power output (and frankly, a compound bow is larger and would probably have more power anyway than a compound crossbow), then this strategy is inferior to just equipping the army with compound bows (where they wouldn't have to reduce their fire rate and all 3 lines could maintain fire/volleys), aside from the fact that it would require MC to also create non-crossbow compound bows. But he already "invented" the mechanism, making a non-crossbow version of it really shouldn't be that hard.
To repeat a point I made in my initial post:
Even if the already-trained archers could use those bows, it doesn't seem like they have the finances to mass-produce both, and the greater benefit is in practically tripling their battle-ready ranged forces. [...] That said, I'm fairly certain if their standing army were larger and could handle the bows he developed, it would be a better option to focus on those, for the reasons stated in the manga.
You're right in that I misidentified the kind of bow he designed, and that he has technically designed what he'd need to make compound bows. However, that doesn't change their financial situation or their bigger issue: their army is apparently only 400 strong, and rereading the last two chapters, almost certainly made of mixed forces (he noted in the last chapter that many are militia, and said militia cannot be put through the training necessary to use bows well enough). There is no technology available to them, or that can be developed, with their resources, knowledge, and time remaining before the conflict, that is going to come anywhere near the value of adding 800 crossbowmen to their combat-ready manpower, even if they're using the cheaper, composite-based crossbows he developed, with only the elites using the compound/composite combination shown off in the chapter. If that 400 soldiers mentioned were
just their territory's archers, I could understand prioritizing compound bow production (assuming they had the finances).
As for training the populace with regular bows, regular archers take significantly more time and investment than you might think -to the degree that the English had a saying, "If you want to train a longbowman, start with his grandfather." In the last chapter, he himself notes that it would take rigorous training and consistent practice -and that's assuming there isn't the need to build up the same (frankly insane) level of musculature needed for a longbow. He has no more than roughly a year and a half to build up the territory's forces, as it doesn't seem to be earlier than spring, and he's got until the beginning of next year's fall before the conflict begins. Unless everyone he gives a bow to has the same level of genius talent with it as his brother does with the sword, it's just not going to be enough time. Meanwhile, training with a crossbow can be done in a few months to 'good enough' levels, and can even be done by people missing an arm, provided they have something like a fallen log or barricade to brace it on (most tools to aid in reloading rely on leg strength, so as long as they can march, they'd still be able to do it).