It is good to remember that not all mounted troops were cavalry.
One way of having a fast-moving infantry force was to issue them horses. In the same way that Mechanized Infantry now rides APCs to where they are needed, the Mounted Infantry rode horses. In parallel, too, Mounted Infantry have to be able fight while both mounted and dismounted.
To add to this parallelism, Mounted Infantry had a "tare" in that one in four was needed to curb and corral the horses while the others dismounted. MechI has a similar issue with drivers and TC in APCs. The collected transportation resource is a tempting target, so it needs defending.
Enough that the fellow manning the curb chains might only have one had free at a time. So, looping him up in lanyards to his pistol magazines was not seen as a detriment. The drill, if I remember my GF's TSG "Guide for Regulating Mounted Infantry" (dated ca 1931, and sadly, long gone loaned out--foolishly--long ago) was that the trooper muffed the grab on the magazine during reload, you could grab the lanyard and haul in, rather than loose the horses being held.
Mounted Infantry were the ones who were to be saddled (NPI) with that awful clamp and lanyard rig for the M1903. If they were engaged while riding, the hasty defense would rely upon pistols until one could wrestle the rifles loose.
Cavalry was expected to fight from horseback, not dismounted. Almost all tools to that end--barring lances--were lashed, lanyard-ed, or otherwise tied off, or they'd be lost.