Also many U.S. cavalry units did become helicopter units at least in the case of the Marine Corps (see the movie We Were Soldiers, I love that movie). Still it does feel a little weird regardless.
Since, on paper, the US Air Force had a monopoly on fixed-wing aircraft, the US Army had to embrace helicopters to have anything of their own in the air...
Of course, that's only one interpretation. In fact, helicopters went through some of the same development cycle that regular aircraft went through: the first helicopters were developed with an eye towards taking over the short-range reconnaissance and artillery-spotting duties that light aircraft had done, just as the first aircraft were regarded as 'eyes in the sky' to take over the recon duties of light cavalry (before WWI).
By the 1950s the helicopter had taken on the roles of medical evacuation (pioneered in the Korean War - MASH was based on a memoir, not fiction!) and short-range 'insertion' of combat troops. The conversion of the US 101st Airborne Division into a helicopter-borne 'Air Mobile' Division and the creation of the 1st Air Cavalry Division, as much as anything, provided a pair of divisional organizations to experiment under fire with everything that could be done with helicopters, in Vietnam. That experience led directly to the provision of helicopter 'gun ships' so that helicopters could both insert troops into the battle and support them with on-hand firepower of all kinds, and get them out again if things went FUBAR.
Point is, Helicopters were certainly an extension of the Light Cavalry, but they also had entirely different capabilities than light cavalry of any kind ever had. That capability is sadly lacking in the game, but since the Support Units in Civ VI have all kinds of variable capabilities, that, it seems to me, is the best place to put Helicopters.
The Light Cavalry line right now jumps from Classical (Horseman) to Industrial (Cavalry) to Atomic (Helicopter) Eras, and people complain about the Classical - Industrial Gap constantly in these Forums. Move the Helicopter to Support and take advantage of that to revise the 'Light Cavalry' as well:
Classical Era:
Horseman as now
Renaissance Era:
Hussars - the most distinctive light cavalry of Europe in the 18th and early 19th centuries.
Modern Era:
Armored Car - a very fast ground unit, but very sensitive to terrain like Rainforest, Woods or Marsh
Information Era:
Stryker - the modern multi-wheeled armored car/armored personnel carrier/weapons platform. Fastest ground unit in the game.
This would both give better functionality to the 'Light Cavalry', and give the game some really striking graphics - Hussars were some of the most flamboyant combat troops that ever saw gun smoke, and mid-20th century armored cars came in a fabulous variety, from 'Dingo' 4-wheeled light types to cannon-bearing 8-wheeled 'heavies'. And, I might add, most of the European Armored Car units kept their Cavalry titles: the 11th Hussars of the British Army was one of their best armored car raiding and recon units in the Western Desert in 1941 - 42, so you wouldn't even have to rename the unit when it gets Upgraded!