The machine gun killed horse cavalry before the semi-automatic and automatic rifle had a chance to (and really the rifled musket and earliest cartridge firing repeaters had horse cavalry on life support well before that). In T2K, horse cavalry are going to be dragoons, not true cavalry, and do their fighting on foot in all but the most permissive environments (note: Absolutely nowhere in the US would be considered a permissive environment. The UK or other places where firearms aren't very plentiful, maybe).
The ideal weapon for CONUS horse cavalry units would probably the M4, but the M16 would be entirely workable and the M16EZ would get the job done. No reason or need for anything more compact than that, and issuing them SMGs would actually tend to be contraindicated by role -- horse cavalry is an asset for patrolling and reconnoitering relatively open terrain where they have a mobility advantage. Even if they are ordered only to fight in the defense and in a pinch, they'd still be wanting weapons that let them engage fully across the 0-3 or 400 meters riflemen can credibly own without specialized equipment or an incredibly obliging enemy.
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