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The seriously odd thing is that the US Army used Sections too in certain cases. I was with the 111th Army Group (Det) and several Engineering units had section listed in their TOE, and actively refereed to them as such.
In old US usage a Section was an intermediate between an Inf Plt and squad, each section having 2 Squads. That has long since been outmoded.
Only if, by "long since" you mean "since the late 1990's" - the Alaska Army National Guard still used sections of 2 squads with assigned Section Sergeants in at least some of the units in the early 90's. It was in phase-out when I entered the army in '87, but was not completely phased out through at least 1994, when members of my gaming group, squad leaders themselves, made mention of their section sergeants.
Note that Traveller has always used:
Fireteam (ø)
Squad (•)
Section (••)
Platoon (•••)
Company (|)
Battalion (||)
Regiment (|||)
Brigade (X)
Division (XX)
Corps (XXX)
Army (XXXX)
Army Group (XXXXX)
It would be nice if the half-company (••••), aka Column, got mentioned as an optional unit as well. At present, it's not used much save for a few third world nations that use 8-10 columns to the battalion, and the two I saw that used that as a non-ad-hoc didn't use companies at all - each column was 8-10 squads of 8-10 men.
The only truly universal levels seem to be the Company of 100-200 men, the squad of 8-20, the Regiment of 2000-5000. And those are the medieval prototypes. (Battalion is derived from medieval as well, being 1/3 of the forces present at a battle for a given side... but could be as low as one company or as high as 30.)
Even then, there's a huge raft of variance in how things get organized - The US had sections as a default standing unit level from the 1820's to the early 1970's, but the fire team as a formal organizational element were post vietnam (but use as a tactical unit dates back earlier); even still, while formalized, it's assignment is usually local decision at the platoon or company level.
In all seriousness - avoiding Britishisms would be a good practice. Use of the much more broadly utilized NATO standards, and the (already canonical for the OTU) US 1970's era hierarchical structure makes for better compatibility with prior editions.