I'd be OK playing in a game that restricted the number of proficiencies, but it does depend a bit on what you define as "trained" or "proficient", and how much training time you alot for a particular training period.
But, realistically, a professional soldier should have wide proficiency even with relatively brief training periods, because you not only need to be able to handle your own weapons, but know the types you're likely to face, how they operate, and how you counter them.
It's not a precise parallel, since modern weapons are generally much easier to learn to employ than most medieval weapons, but as a point of comparison here's what I was trained to employ during just my first four years in the military:
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Qualified "Expert" (as in: was formally tested in some form of combat training course and received a score in the highest category, equating to greater combat proficiency): Pistols (M9), Carbines/Rifles (M16), Medium Machine Guns (M60, M240), Heavy Machine Guns (M2), Grenades (M67), Bradley Fighting Vehicle, Abrams Tank
Qualified (as in: was formally tested in some form of combat training course and received a "qualifying" score, equating to basic combat proficiency): Carbines/Rifles (M4), Heavy Machine Guns (Mk19), TOW missile
Trained (as in: received formal training in the use, maintenance, and operation, to include live fire operation, but did not have a combat course graduating or qualifying event -- enabling basic proficiency): Pistols (Makarov, M1911, Browning Hi-Power), SMGs (M3, MP5, Danish Madsen, Swedish K), Carbines/Rifles (AK47, AK74, SKS, FN/FAL, G3), Light/Medium Machine Guns (M249, PKM, MG3), Heavy Machine Guns (DShK), Demolitions (C4, TNT), Mines (M16, M18, M21), Recoilless Rifles (90mm, 106mm), Mortars (60mm, 81mm)
Familiarized (as in: received a basic orientation on operation sufficient to be safe, possibly with some live fire operation): too many to list, but includes things like the Stinger missile, M60 tank, M101 105mm howitzer, and others.
I wouldn't claim mastery of any of these weapons with the possible exception of the Bradley Fighting Vehicle and Abrams Tank -- but it illustrates the number of proficiencies that a modern professional soldier can pick up in a relatively short period of time.
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