Why does the Ignition button reset Steel Battalion?
With some research online, I was able to find a quick and painless solution. I'm posting the question and my answer not only to share my experience with a broader audience, but to allow for some other user with a better answer to come along as well.
It appears to be tied to the Xbox's modded firmware allowing for a soft reset. The original Xbox does not have much to do on the dashboard and did not come with a native reset; the developers of the firmware built this feature in, allowing for users to easily return to their more robust dashboard. Unfortunately, the Ignition button maps to the button combination for the reset. Whether this was intended by Capcom or not remains a mystery to me.
Posts on various forums as far back as 2004 indicate that pushing all the way forward on the right stick while pressing the Ignition button will bypass the mod's reset function. The problem and solution are even mentioned on the current version of the game's Wikipedia article, despite that probably not being the place for that information. I was able to confirm this works and proceed with the first three missions in the game with no further reset problems.