While I admire the way you think that's a bit impractical for a number of reasons. In most, even 99% of cases, the issued weapon is the M16. Only medics (the Navy has corpsmen) and in some cases, aviators, tankers and some officers (staff usually) are issued sidearms with which they have to qualify. It's nearly impossible to lug an M16 around daily especially to the office. Imagine leaving your rifle in the latrine (the navy calls it the Head and civilians call it the bathroom) where any janitor could find it.
I was an officer in an infantry outfit. Guys like me lead the killers in their duties of knocking off the enemy. Don't tell me I don't "need a sidearm!" I was also Executive Officer of the post pistol team (second in command) for a time. Back in those days, when General Dozier was kidnapped over in Italy by terrorists (later executed) my bosses who were the Division Commander and his key staff ordered all division staff OFFICERS to come to work after drawing a sidearm (1911 45acp) and at least two magazines of ball ammo and wear said sidearm under the duty uniform, concealed. When the division HQ building was hit with a bomb threat, they evacuated and then sent all of us LIEUTENANTS back inside to look for the "device!" It sure made me feel special, I'll tell you. Special and decidedly expendable.
(...OK, just kidding everyone...calm down...)
As a veteran Im aware of the issues you note. My position is that if we can afford to bomb aspirin factories with million dollar bombs and if we can afford to execute "very small" strikes against Syria with cruise missiles costing 10s of millions then there is enough money laying around to to have both available to our troops along with the money necessary for them to qualify on more than one weapon. True, most troops simply don't require M16s. Then again, Im not the type to try to drive nails with a screwdriver. Different tools for different tasks. A sidearm isn't going to get in the way of latrine duty, weed n seed, or stirring oatmeal in the mess. I am proposing either/or for the majority of service members, most getting sidearms. Infantry, security, and similar troops should ALSO have a sidearm available along with their M16. Im aware of the issue of the weight of the total combat gear that infantry soldiers struggle with, they should still have the option to carry both.
As far as losing the weapon...all I have to say is that 18 year old Israeli girls seem to be able to find a place in their bikinis for their compacts during a day at the beach with their friends and still remember to not lose their rifle. I cant imagine why properly trained American troops on base shouldnt somehow be able to pull off a task as simple as not losing a sidearm in a latrine. (Place a hook near the handle for hanging the belt so that they have to reach past it to open the door. Im sure there is a contractor out there that should be able to produce them for only $37,000 per unit.)
Unless you were looking for "a section 8", I doubt you were standing up front using your .45 as a pointer for your unit to follow. I suspect the unfortunate reality is that he day an infantry officer "needs" to personally shoot someone in the field there should be plenty of M16s laying around for them to use. Maybe a marksman such as yourself should have been issued something more capable than the M16 all along, yet still have had the option to carry a sidearm. The rest of the days in your career you should have had access to a sidearm instead.