It's interesting that this argument has been used on both sides of this debate -- here and in the 3/7 patch thread. From my pov, the barracks buff pretty much means everyone should be building barracks right away (as opposed to eventually because they are a core building). I saw that as too much homogeneity.
If they're nerfed Science-wise, then warmongerers will not be able to afford rushing Swordsmen/Water mill techs without falling behind - and tbh it already is barely affordable even with some free ruin techs unless you also get enough Copper/Iron tiles to at least give you some other gain than just Swordsmen. Even Military Theory will be hard to get before Libraries In addition, warmongerers will have yet another problem posed before them - what do you build first after Monuments/Shrines? Barracks won't be enough for Science without having Science (or with only 1) but without them you cannot field units, so you'd need to spend way more time on Buildings before and after units. Libraries would just take over the spot of Barracks as what everyone builds ASAP, all the while warmongering would get yet another barrage of nerfs. It'd be better if Walls got better at Supply while Barracks got worse, though even that'd be a nerf as you'd need to build up your defenses (spend more production on buildings) before sending out an assault force, whereas you're likely the one who gets Markets later when even Markets might not be enough for your needs.
If Barracks had no Science or just 1, then warmongerer would need to build Council, Library ASAP anyway (meaning his build order would be similar to that of a peaceful guy, but he'd absolutely need Barracks for supply), and tech costs would need further trimming down due to less Science overall while teching is already kind of slow. Authority would need to be buffed for sure if Barracks got nerfed, so maybe remove 1 Science from Barracks, but add +1 Science per Barrack in Authority so warmongerers aren't hurt by needing to play exactly the same as Progress/Tradition, but peacemongers build Libraries before them anyway unless they need supply? Or buff Authority in some other way so you won't fall off hard if you don't rush Libraries every time. It'd still probably require early tech costs reduction of some sort.
A new thread would need to be opened for that I think.
If we roughly agree on what we're polling!
In my mind, it's the necessity of hurrying to build barracks in most games, as a result of the cap. My guess is that the science aspect is meant to justify that necessity -- but nerfing the science wouldn't necessarily be what we'd be polling. You'd still probably have to prioritize barracks, even if they lost the beakers.
My own preference would be for the barracks to lose both the +1 to the cap and (less so) +1 science... then make up the unit losses by scaling the cap upward in the early game.
After those changes, Barracks would be so awful even warmongerers wouldn't bother to get them. I'm a warmongering menace but after your changes, I'd much rather build a Council over them as it'd produce more Science (+birth Science), be on the way to mandatory Libraries, it'd cost no maintenance and early XP is not worth having 1 unit less as barracks cost enough to get you more than one horseman, whereas the percentage supply it has is not worth anything for someone who has less pop by definition as he delays buildings to get units and his policies are outright worse at Food/growing. The only people who'd ever see Armouries appear in their can-be-built scroll would be the Japanese as the Dojo doesn't require Barracks, or those who plan to go Imperialism (which is not exactly the best of its kind) and are somehow close enough or simply really have nothing else to do as they've built all the other buildings (excl. Stone Works with 1 Quarry I guess, they're worse) incl. Arenas so they got the Barracks.
I wouldn't enjoy your changes forcing me to go Pottery - Wheel/Trapping - Wheel/Trapping - Trade - Calendar - Writing every game or fall behind in everything as Barracks, instead of producing only 2.5 times Science (because you have to work Scientist specialists every time anyway) less, would be 5 times worse at Science than Libraries or more if you consider the imminent Scientist. It'd probably make Authority unplayable as you either would have to delay your conquest until you not only get Libraries and build armies while hoping your supply cap while you are on a tree that has awful Food is enough, or you'd delay Libraries and enjoy seeing people on Philosophy while you've almost gotten the Forge tech, especially after the killed units yields were reduced by IIRC 20%. That means the policy tree reliant on balling wouldn't be able to ball.
You'd definitely make one thing homogenous after this change - neither warmongerers nor peaceful would build Barracks until it was the last/second to last (stone works 1 stone/marble) building available. Tell me honestly, when would you be building those... "balanced" Barracks? Because I think dead last is the answer as the XP gain just isn't worth one produced unit when early game AIs like to spam units and it's easier to just hit them a bit, you also suggested more generous scaling supply cap which could potentially take the function of the entire building chain over, 1 Science for 1 Gold is not worthwhile when early game dipping in negative Gold can happen often despite worked Markets and little roads because of how bad city connections are, meaning it'd have no real positives at all until after Arenas.
How much "rush to barracks" are we talking about. Does that mean building barracks before:
Shrine
Monument
Well/water mill
Council
Military units
Indont think anyone is questioning its a necessary building, but is it so strong it trumps all other buildings and units...or is it simply a strong essential building in a sea of strong essential buildings?
I agree, I always build Water Mill/Shrine/Monument first. Not Well unless I have Currency or at least work Markets everywhere though, it's just not worth 2 Maintenance when Gold is so scarce and other alternatives are superior.