Actually you do get a choice of who your general is, whether it's in multiplayer, custom battle, or campaign.
While sometimes you may get stuck with an Ashigaru general, you can always remove him from your army (in campaign mode) and then the next unit with the highest honor gets the spot. This sometimes won't matter if your army is mostly made up of ashigaru, though, or similar cheap troops like archers. In multiplayer and custom, the first unit you purchase is always the general.
The problem with removing a general with high honor and yet useless ability (musket generals, which are extremely rare, but serve a good enough purpose in this example) is you lose the morale and honor bonuses he'd give your army even if he wasn't actually doing any fighting during the next battle.