I do have what is maybe an odd question...since I have only ever had gold fish, I only have that to draw on. Tetras seem to wait for the food to fall to eat it mid fall? I thought all fish came to the top to eat?
A bit of food escaped to the bottom without being fed. Would something like a shrimp take care of that?
You might have noticed on your fish purchasing/viewing adventures that different fish have different shapes to their mouths. As the image below will illustrate, these mouth shapes aid in feeding from different levels of the water and food availability. Tetras, as you mentioned, would prefer to wait until the food reaches the bottom as their mouths are forward facing and they are fast swimmers. Other fish like gouramis, bettas, and the common livebearers like mollies, swordtails and guppies will tend to eat the food from the surface as their mouths are upturned (if even slightly) (also, with the labyrinth fish like gouramis and bettas, the mouth shape is more aimed towards aiding in breathing from air, but it also affects their eating habits). Down turned mouths like those found on ottos, plecos, and the clown loach as depicted will tend to eat food from the bottom of the aquarium.
I have found through my experience that if a fish like a tetra is not fed at the mid level and is only offered food at the surface or the bottom, they will change their habits, but they seem to greatly prefer the normal food availability they are accustomed to. I know my tetras will gladly eat the pellets I feed that sink to the bottom, as will my guppies, but I notice that they do so much more after all of the food at the surface and mid level has been eaten.
A shrimp will help in eating the food that makes its way to the bottom, yes, and they are very proficient at it. You rarely have to add extra food for the shrimp specifically- if anything falls, they'll find it.