I noticed this starter in a cased c.1810 flintlock.
It looks like a mallet, but obviously is not intended to be used as a mallet. The rod section is not a handle, it would break if you used this as a mallet.
[edit: re-reading the quote below, I think maybe the ball was first tapped with the end of the barrel-shaped handle, not by slapping the top, but by using the rod as a handle. So it is a "mallet" but not one you really whack things with.]
This has to be a loading "mallet" as named in old texts. And it surely was called a "mallet" only because it resembled a mallet.
To give a parallel, seamen used a lever called a "serving mallet" to seal rope by wrapping it tightly with small cord. See photo. You don't hit anything with a serving mallet. It is purely a lever that resembles a mallet. So they called it a mallet.
[corrected] Sir Michael posted this a while back:
[block quote]
"The British Army Rifle Regiment were equiped with what was referred to as a loading mallet. However, no detail descriptions or instructions for use have ever been found.
"Along this same line, I found this regarding a similar device used by the U.S. Army:
'Practical Instructions for Military Officers, for the District of Massachusetts, published 1811
'Equipment
'The balls attached to the cartridges are enclosed in a linen or milled leather patch well saturated with grease; when the powder is emptied into the rifle, the ball is to be separated from the paper, placed upon the muzzle and driven in with a stroke of the mallet, as will be described in the exercise of the rifle. . . .
'V. Load! One Compound Motion
Turn up the right hand and shake the powder into the barrel, pressing the cartridge with the thumb and finger, to force out the powder; instantly bring the paper to the mouth and with the teeth separate it from the ball and, patch, which place upon the muzzle, the stitched side up, and instantly slide the left hand to the muzzle and place the fore finger upon the ball; at the same time, with the right hand, grasp the mallet, draw it partly out, and seize the handle.
'VI. Drive Ball! One Compound Motion
Bring up the mallet, flipping the finger from the ball, and with one or two strokes drive the ball into the muzzle; with a quick motion, place the end of the handle upon the ball and grasp it with the thumb and finger of the left hand, and with a few smart strokes upon the mallet with the right hand, drive the ball down the full length of the handle; instantly return the mallet to its sheath and seize the ramrod with the thumb and finger of the same hand, the thumb up. "
[end block quote]
I think the starter in the photo is original to the set, that it was called a mallet because it resembled a mallet, and that it was used exactly as "short starters" today are used.